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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="bhsp_1.html#NM1">RealUnitarySpace</a>;<br/>

<b>set </b><font color="Maroon">c<sub>2</sub></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">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="rusub_2.html#K5" target="_self">SubJoin</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="rusub_2.html#K6" target="_self">SubMeet</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #);<br/>
<a NAME="E1:75"/><i><font color="Green">E55</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <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="rusub_2.html#K3" target="_self">Subspaces</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="rusub_2.html#K5" target="_self">SubJoin</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="rusub_2.html#K6" target="_self">SubMeet</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #) holds <font color="Olive">b<sub>1</sub></font> <a href="lattices.html#K1">"\/"</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> <a href="lattices.html#K1">"\/"</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="75_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</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="rusub_2.html#K3" target="_self">Subspaces</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="rusub_2.html#K5" target="_self">SubJoin</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="rusub_2.html#K6" target="_self">SubMeet</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #);<br/>

<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>4</sub></font> as    <a href="rusub_1.html#M1">Subspace</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="rusub_2.html#D3" target="_self">Def3</a></i>;<br/>
<b>thus </b><font color="Maroon">c<sub>3</sub></font> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">c<sub>4</sub></font> = 
<span class="p1">(<span class="default"><a href="rusub_2.html#K5" target="_self">SubJoin</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="ref" href="lattices.html#D1">LATTICES:def 1</a></i>
<br/>.= 
<font color="Maroon">c<sub>5</sub></font> <a href="rusub_2.html#K1" target="_self">+</a> <font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="ref" href="rusub_2.html#D7" target="_self">Def7</a></i>
<br/>.= 
<font color="Maroon">c<sub>6</sub></font> <a href="rusub_2.html#K1" target="_self">+</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="txt" href="rusub_2.html#E6"><i><font color="Green">Lemma6</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rusub_2.html#K5" target="_self">SubJoin</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="ref" href="rusub_2.html#D7" target="_self">Def7</a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="ref" href="lattices.html#D1">LATTICES:def 1</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E2:75"/><i><font color="Green">E56</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</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="rusub_2.html#K3" target="_self">Subspaces</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="rusub_2.html#K5" target="_self">SubJoin</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="rusub_2.html#K6" target="_self">SubMeet</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #) holds <font color="Olive">b<sub>1</sub></font> <a href="lattices.html#K1">"\/"</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="lattices.html#K1">"\/"</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="lattices.html#K1">"\/"</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="lattices.html#K1">"\/"</a> <font color="Olive">b<sub>3</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="75_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font>, <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="rusub_2.html#K3" target="_self">Subspaces</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="rusub_2.html#K5" target="_self">SubJoin</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="rusub_2.html#K6" target="_self">SubMeet</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #);<br/>

<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>5</sub></font> as    <a href="rusub_1.html#M1">Subspace</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="rusub_2.html#D3" target="_self">Def3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>6</sub></font> <a href="rusub_2.html#K1" target="_self">+</a> <font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>7</sub></font> <a href="rusub_2.html#K1" target="_self">+</a> <font color="Maroon">c<sub>8</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="rusub_2.html#K3" target="_self">Subspaces</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="rusub_2.html#K5" target="_self">SubJoin</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="rusub_2.html#K6" target="_self">SubMeet</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #) <b>by </b><i><a class="ref" href="rusub_2.html#D3" target="_self">Def3</a></i>;<br/>
<b>thus </b><font color="Maroon">c<sub>3</sub></font> <a href="lattices.html#K1">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><a href="rusub_2.html#K5" target="_self">SubJoin</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">c<sub>5</sub></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">c<sub>1</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font>,<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">c<sub>1</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></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">c<sub>1</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>10</sub></font>
<b>by </b><i><a class="ref" href="rusub_2.html#D7" target="_self">Def7</a></i>
<br/>.= 
<font color="Maroon">c<sub>6</sub></font> <a href="rusub_2.html#K1" target="_self">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="rusub_2.html#K1" target="_self">+</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="rusub_2.html#D7" target="_self">Def7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="rusub_2.html#K1" target="_self">+</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="rusub_2.html#K1" target="_self">+</a> <font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="ref" href="rusub_2.html#T6" target="_self">Th6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rusub_2.html#K5" target="_self">SubJoin</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="ref" href="rusub_2.html#D7" target="_self">Def7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rusub_2.html#K5" target="_self">SubJoin</a> <font color="Maroon">c<sub>1</sub></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">c<sub>1</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="ref" href="rusub_2.html#D7" target="_self">Def7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rusub_2.html#K5" target="_self">SubJoin</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>5</sub></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">c<sub>3</sub></font> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="ref" href="lattices.html#D1">LATTICES:def 1</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E3:75"/><i><font color="Green">E57</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <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="rusub_2.html#K3" target="_self">Subspaces</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="rusub_2.html#K5" target="_self">SubJoin</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="rusub_2.html#K6" target="_self">SubMeet</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #) holds <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="lattices.html#K2">"/\"</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="lattices.html#K1">"\/"</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="75_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</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="rusub_2.html#K3" target="_self">Subspaces</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="rusub_2.html#K5" target="_self">SubJoin</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="rusub_2.html#K6" target="_self">SubMeet</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #);<br/>

<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>4</sub></font> as  <a href="bhsp_1.html#V1">strict</a>  <a href="rusub_1.html#M1">Subspace</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="rusub_2.html#D3" target="_self">Def3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>5</sub></font> <a href="rusub_2.html#K2" target="_self">/\</a> <font color="Maroon">c<sub>6</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="rusub_2.html#K3" target="_self">Subspaces</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="rusub_2.html#K5" target="_self">SubJoin</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="rusub_2.html#K6" target="_self">SubMeet</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #) <b>by </b><i><a class="ref" href="rusub_2.html#D3" target="_self">Def3</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">c<sub>4</sub></font> = 
<span class="p1">(<span class="default"><a href="rusub_2.html#K5" target="_self">SubJoin</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></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_2.html#K5" target="_self">SubJoin</a> <font color="Maroon">c<sub>1</sub></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#K6" target="_self">SubMeet</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>,<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/>.= 
<span class="p1">(<span class="default"><a href="rusub_2.html#K5" target="_self">SubJoin</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="ref" href="rusub_2.html#D8" target="_self">Def8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="rusub_2.html#K2" target="_self">/\</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="rusub_2.html#K1" target="_self">+</a> <font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="ref" href="rusub_2.html#D7" target="_self">Def7</a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="ref" href="rusub_2.html#T23" target="_self">Th23</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E4:75"/><i><font color="Green">E58</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <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="rusub_2.html#K3" target="_self">Subspaces</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="rusub_2.html#K5" target="_self">SubJoin</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="rusub_2.html#K6" target="_self">SubMeet</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #) holds <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>2</sub></font> <a href="lattices.html#K2">"/\"</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="75_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</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="rusub_2.html#K3" target="_self">Subspaces</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="rusub_2.html#K5" target="_self">SubJoin</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="rusub_2.html#K6" target="_self">SubMeet</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #);<br/>

<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>4</sub></font> as    <a href="rusub_1.html#M1">Subspace</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="rusub_2.html#D3" target="_self">Def3</a></i>;<br/>
<b>thus </b><font color="Maroon">c<sub>3</sub></font> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">c<sub>4</sub></font> = 
<span class="p1">(<span class="default"><a href="rusub_2.html#K6" target="_self">SubMeet</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>3</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>5</sub></font> <a href="rusub_2.html#K2" target="_self">/\</a> <font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="ref" href="rusub_2.html#D8" target="_self">Def8</a></i>
<br/>.= 
<font color="Maroon">c<sub>6</sub></font> <a href="rusub_2.html#K2" target="_self">/\</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="ref" href="rusub_2.html#T14" target="_self">Th14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rusub_2.html#K6" target="_self">SubMeet</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="ref" href="rusub_2.html#D8" target="_self">Def8</a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="ref" href="lattices.html#D2">LATTICES:def 2</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E5:75"/><i><font color="Green">E59</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</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="rusub_2.html#K3" target="_self">Subspaces</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="rusub_2.html#K5" target="_self">SubJoin</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="rusub_2.html#K6" target="_self">SubMeet</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #) holds <font color="Olive">b<sub>1</sub></font> <a href="lattices.html#K2">"/\"</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="lattices.html#K2">"/\"</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="lattices.html#K2">"/\"</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="lattices.html#K2">"/\"</a> <font color="Olive">b<sub>3</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="75_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font>, <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="rusub_2.html#K3" target="_self">Subspaces</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="rusub_2.html#K5" target="_self">SubJoin</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="rusub_2.html#K6" target="_self">SubMeet</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #);<br/>

<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>5</sub></font> as    <a href="rusub_1.html#M1">Subspace</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="rusub_2.html#D3" target="_self">Def3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>6</sub></font> <a href="rusub_2.html#K2" target="_self">/\</a> <font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>7</sub></font> <a href="rusub_2.html#K2" target="_self">/\</a> <font color="Maroon">c<sub>8</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="rusub_2.html#K3" target="_self">Subspaces</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="rusub_2.html#K5" target="_self">SubJoin</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="rusub_2.html#K6" target="_self">SubMeet</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #) <b>by </b><i><a class="ref" href="rusub_2.html#D3" target="_self">Def3</a></i>;<br/>
<b>thus </b><font color="Maroon">c<sub>3</sub></font> <a href="lattices.html#K2">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><a href="rusub_2.html#K6" target="_self">SubMeet</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">c<sub>5</sub></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#K6" target="_self">SubMeet</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></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">c<sub>1</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></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#K6" target="_self">SubMeet</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>10</sub></font>
<b>by </b><i><a class="ref" href="rusub_2.html#D8" target="_self">Def8</a></i>
<br/>.= 
<font color="Maroon">c<sub>6</sub></font> <a href="rusub_2.html#K2" target="_self">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="rusub_2.html#K2" target="_self">/\</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="rusub_2.html#D8" target="_self">Def8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="rusub_2.html#K2" target="_self">/\</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="rusub_2.html#K2" target="_self">/\</a> <font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="ref" href="rusub_2.html#T15" target="_self">Th15</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rusub_2.html#K6" target="_self">SubMeet</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="ref" href="rusub_2.html#D8" target="_self">Def8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rusub_2.html#K6" target="_self">SubMeet</a> <font color="Maroon">c<sub>1</sub></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#K6" target="_self">SubMeet</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="ref" href="rusub_2.html#D8" target="_self">Def8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rusub_2.html#K6" target="_self">SubMeet</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<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"><font color="Maroon">c<sub>3</sub></font> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="lattices.html#K2">"/\"</a> <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/>


</div><b>end;</b></div>
<a NAME="E6:75"/>
for <font color="Olive">b<sub>1</sub></font>, <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="rusub_2.html#K3" target="_self">Subspaces</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="rusub_2.html#K5" target="_self">SubJoin</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="rusub_2.html#K6" target="_self">SubMeet</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #) holds <font color="Olive">b<sub>1</sub></font> <a href="lattices.html#K2">"/\"</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="lattices.html#K1">"\/"</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="75_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</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="rusub_2.html#K3" target="_self">Subspaces</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="rusub_2.html#K5" target="_self">SubJoin</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="rusub_2.html#K6" target="_self">SubMeet</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #);<br/>

<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>4</sub></font> as  <a href="bhsp_1.html#V1">strict</a>  <a href="rusub_1.html#M1">Subspace</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="rusub_2.html#D3" target="_self">Def3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>5</sub></font> <a href="rusub_2.html#K1" target="_self">+</a> <font color="Maroon">c<sub>6</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="rusub_2.html#K3" target="_self">Subspaces</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="rusub_2.html#K5" target="_self">SubJoin</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="rusub_2.html#K6" target="_self">SubMeet</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #) <b>by </b><i><a class="ref" href="rusub_2.html#D3" target="_self">Def3</a></i>;<br/>
<b>thus </b><font color="Maroon">c<sub>3</sub></font> <a href="lattices.html#K2">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><a href="rusub_2.html#K6" target="_self">SubMeet</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">c<sub>4</sub></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#K6" target="_self">SubMeet</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font>,<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">c<sub>1</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></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#K6" target="_self">SubMeet</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="ref" href="rusub_2.html#D7" target="_self">Def7</a></i>
<br/>.= 
<font color="Maroon">c<sub>5</sub></font> <a href="rusub_2.html#K2" target="_self">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="rusub_2.html#K1" target="_self">+</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="rusub_2.html#D8" target="_self">Def8</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="ref" href="rusub_2.html#T24" target="_self">Th24</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E7:75"/><b>then </b>
(  <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">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="rusub_2.html#K5" target="_self">SubJoin</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="rusub_2.html#K6" target="_self">SubMeet</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #) is <a href="lattices.html#V4">join-commutative</a> &amp;  <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">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="rusub_2.html#K5" target="_self">SubJoin</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="rusub_2.html#K6" target="_self">SubMeet</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #) is <a href="lattices.html#V5">join-associative</a> &amp;  <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">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="rusub_2.html#K5" target="_self">SubJoin</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="rusub_2.html#K6" target="_self">SubMeet</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #) is <a href="lattices.html#V8">meet-absorbing</a> &amp;  <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">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="rusub_2.html#K5" target="_self">SubJoin</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="rusub_2.html#K6" target="_self">SubMeet</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #) is <a href="lattices.html#V6">meet-commutative</a> &amp;  <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">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="rusub_2.html#K5" target="_self">SubJoin</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="rusub_2.html#K6" target="_self">SubMeet</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #) is <a href="lattices.html#V7">meet-associative</a> &amp;  <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">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="rusub_2.html#K5" target="_self">SubJoin</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="rusub_2.html#K6" target="_self">SubMeet</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #) is <a href="lattices.html#V9">join-absorbing</a> )
 
<b>by </b><i><a class="txt" href="rusub_2.html#E1:75"><i><font color="Green">E55</font></i></a>, <a class="txt" href="rusub_2.html#E2:75"><i><font color="Green">E56</font></i></a>, <a class="txt" href="rusub_2.html#E3:75"><i><font color="Green">E57</font></i></a>, <a class="txt" href="rusub_2.html#E4:75"><i><font color="Green">E58</font></i></a>, <a class="txt" href="rusub_2.html#E5:75"><i><font color="Green">E59</font></i></a>, <a class="ref" href="lattices.html#D4">LATTICES:def 4</a>, <a class="ref" href="lattices.html#D5">LATTICES:def 5</a>, <a class="ref" href="lattices.html#D6">LATTICES:def 6</a>, <a class="ref" href="lattices.html#D7">LATTICES:def 7</a>, <a class="ref" href="lattices.html#D8">LATTICES:def 8</a>, <a class="ref" href="lattices.html#D9">LATTICES:def 9</a></i>;<br/>
<b>hence </b><a NAME="E8:75"/>
 <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">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="rusub_2.html#K5" target="_self">SubJoin</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="rusub_2.html#K6" target="_self">SubMeet</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #) is   <a href="lattices.html#NM1">Lattice</a>
 <b>by </b><i><a class="ref" href="lattices.html#D10">LATTICES:def 10</a></i>;<br/>


</div>
