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

<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="msualg_2.html#K14" target="_self">MSSub</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_2.html#K15" target="_self">MSAlg_join</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_2.html#K16" target="_self">MSAlg_meet</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> #);<br/>
<a NAME="E1:69_1_1"/><i><font color="Green">E51</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="msualg_2.html#K14" target="_self">MSSub</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_2.html#K15" target="_self">MSAlg_join</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_2.html#K16" target="_self">MSAlg_meet</a> <font color="Maroon">c<sub>2</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="69_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><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="msualg_2.html#K14" target="_self">MSSub</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_2.html#K15" target="_self">MSAlg_join</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_2.html#K16" target="_self">MSAlg_meet</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> #);<br/>

<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>5</sub></font> as    <a href="subset_1.html#M1">Element</a> of  <a href="msualg_2.html#K14" target="_self">MSSub</a> <font color="Maroon">c<sub>2</sub></font> ;<br/>
<a NAME="E2:69_1_1_1"/><i><font color="Green">E52</font></i>: 
 <a href="msualg_2.html#K15" target="_self">MSAlg_join</a> <font color="Maroon">c<sub>2</sub></font> is <a href="binop_1.html#V1">commutative</a>
 
<b>by </b><i><a class="ref" href="msualg_2.html#T30" target="_self">Th30</a></i>;<br/>
<b>thus </b><font color="Maroon">c<sub>4</sub></font> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">c<sub>5</sub></font> = 
<span class="p1">(<span class="default"><a href="msualg_2.html#K15" target="_self">MSAlg_join</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>

<br/>.= 
the <a href="lattices.html#U2">L_join</a> of <a href="lattices.html#G3">LattStr</a>(# <span class="p1">(<span class="default"><a href="msualg_2.html#K14" target="_self">MSSub</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_2.html#K15" target="_self">MSAlg_join</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_2.html#K16" target="_self">MSAlg_meet</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> #) <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="msualg_2.html#E2:69_1_1_1"><i><font color="Green">E52</font></i></a>, <a class="ref" href="binop_1.html#D2">BINOP_1:def 2</a></i>
<br/>.= 
<font color="Maroon">c<sub>5</sub></font> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">c<sub>4</sub></font>

;<br/>


</div><b>end;</b></div>
<a NAME="E2:69_1_1"/><i><font color="Green">E52</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="msualg_2.html#K14" target="_self">MSSub</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_2.html#K15" target="_self">MSAlg_join</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_2.html#K16" target="_self">MSAlg_meet</a> <font color="Maroon">c<sub>2</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="69_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</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="msualg_2.html#K14" target="_self">MSSub</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_2.html#K15" target="_self">MSAlg_join</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_2.html#K16" target="_self">MSAlg_meet</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> #);<br/>

<b>reconsider </b><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>, <font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>6</sub></font> as    <a href="subset_1.html#M1">Element</a> of  <a href="msualg_2.html#K14" target="_self">MSSub</a> <font color="Maroon">c<sub>2</sub></font> ;<br/>
<a NAME="E2:69_1_1_2"/><i><font color="Green">E53</font></i>: 
 <a href="msualg_2.html#K15" target="_self">MSAlg_join</a> <font color="Maroon">c<sub>2</sub></font> is <a href="binop_1.html#V2">associative</a>
 
<b>by </b><i><a class="ref" href="msualg_2.html#T31" target="_self">Th31</a></i>;<br/>
<b>thus </b><font color="Maroon">c<sub>4</sub></font> <a href="lattices.html#K1">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> = 
the <a href="lattices.html#U2">L_join</a> of <a href="lattices.html#G3">LattStr</a>(# <span class="p1">(<span class="default"><a href="msualg_2.html#K14" target="_self">MSSub</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_2.html#K15" target="_self">MSAlg_join</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_2.html#K16" target="_self">MSAlg_meet</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> #) <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><a href="msualg_2.html#K15" target="_self">MSAlg_join</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_2.html#K15" target="_self">MSAlg_join</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font></span>)</span>

<br/>.= 
the <a href="lattices.html#U2">L_join</a> of <a href="lattices.html#G3">LattStr</a>(# <span class="p1">(<span class="default"><a href="msualg_2.html#K14" target="_self">MSSub</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_2.html#K15" target="_self">MSAlg_join</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_2.html#K16" target="_self">MSAlg_meet</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> #) <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default">the <a href="lattices.html#U2">L_join</a> of <a href="lattices.html#G3">LattStr</a>(# <span class="p1">(<span class="default"><a href="msualg_2.html#K14" target="_self">MSSub</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_2.html#K15" target="_self">MSAlg_join</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_2.html#K16" target="_self">MSAlg_meet</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> #) <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="txt" href="msualg_2.html#E2:69_1_1_2"><i><font color="Green">E53</font></i></a>, <a class="ref" href="binop_1.html#D3">BINOP_1:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default">the <a href="lattices.html#U2">L_join</a> of <a href="lattices.html#G3">LattStr</a>(# <span class="p1">(<span class="default"><a href="msualg_2.html#K14" target="_self">MSSub</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_2.html#K15" target="_self">MSAlg_join</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_2.html#K16" target="_self">MSAlg_meet</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> #) <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">c<sub>6</sub></font>

<br/>.= 
<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> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">c<sub>6</sub></font>

;<br/>


</div><b>end;</b></div>
<a NAME="E3:69_1_1"/><i><font color="Green">E53</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="msualg_2.html#K14" target="_self">MSSub</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_2.html#K15" target="_self">MSAlg_join</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_2.html#K16" target="_self">MSAlg_meet</a> <font color="Maroon">c<sub>2</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="69_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><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="msualg_2.html#K14" target="_self">MSSub</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_2.html#K15" target="_self">MSAlg_join</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_2.html#K16" target="_self">MSAlg_meet</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> #);<br/>

<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>5</sub></font> as    <a href="subset_1.html#M1">Element</a> of  <a href="msualg_2.html#K14" target="_self">MSSub</a> <font color="Maroon">c<sub>2</sub></font> ;<br/>
<a NAME="E2:69_1_1_3"/><i><font color="Green">E54</font></i>: 
<span class="p1">(<span class="default"><a href="msualg_2.html#K15" target="_self">MSAlg_join</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_2.html#K16" target="_self">MSAlg_meet</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_1_3_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>7</sub></font> as  <a href="msualg_1.html#V4">strict</a>  <a href="msualg_2.html#M1" target="_self">MSSubAlgebra</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="ref" href="msualg_2.html#D20" target="_self">Def20</a></i>;<br/>
<a NAME="E2:69_1_1_3_1"/>
<span class="p1">(<span class="default"><a href="msualg_2.html#K16" target="_self">MSAlg_meet</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="msualg_2.html#K11" target="_self">/\</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="ref" href="msualg_2.html#D22" target="_self">Def22</a></i>;<br/>
<b>hence </b><span class="p1">(<span class="default"><a href="msualg_2.html#K15" target="_self">MSAlg_join</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_2.html#K16" target="_self">MSAlg_meet</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>7</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="msualg_2.html#K11" target="_self">/\</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="msualg_2.html#K13" target="_self">"\/"</a> <font color="Maroon">c<sub>9</sub></font>
<b>by </b><i><a class="ref" href="msualg_2.html#D21" target="_self">Def21</a></i>
<br/>.= 
<font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="ref" href="msualg_2.html#T29" target="_self">Th29</a></i>
;<br/><br/>


</div><b>end;</b></div>
<b>thus </b><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> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">c<sub>5</sub></font> = 
the <a href="lattices.html#U2">L_join</a> of <a href="lattices.html#G3">LattStr</a>(# <span class="p1">(<span class="default"><a href="msualg_2.html#K14" target="_self">MSSub</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_2.html#K15" target="_self">MSAlg_join</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_2.html#K16" target="_self">MSAlg_meet</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> #) <a href="binop_1.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>,<font color="Maroon">c<sub>5</sub></font>

<br/>.= 
<font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="txt" href="msualg_2.html#E2:69_1_1_3"><i><font color="Green">E54</font></i></a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E4:69_1_1"/><i><font color="Green">E54</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="msualg_2.html#K14" target="_self">MSSub</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_2.html#K15" target="_self">MSAlg_join</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_2.html#K16" target="_self">MSAlg_meet</a> <font color="Maroon">c<sub>2</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="69_1_1_4"><b>proof </b></a><div class="add">

<b>let </b><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="msualg_2.html#K14" target="_self">MSSub</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_2.html#K15" target="_self">MSAlg_join</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_2.html#K16" target="_self">MSAlg_meet</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> #);<br/>

<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>5</sub></font> as    <a href="subset_1.html#M1">Element</a> of  <a href="msualg_2.html#K14" target="_self">MSSub</a> <font color="Maroon">c<sub>2</sub></font> ;<br/>
<a NAME="E2:69_1_1_4"/><i><font color="Green">E55</font></i>: 
 <a href="msualg_2.html#K16" target="_self">MSAlg_meet</a> <font color="Maroon">c<sub>2</sub></font> is <a href="binop_1.html#V1">commutative</a>
 
<b>by </b><i><a class="ref" href="msualg_2.html#T32" target="_self">Th32</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="msualg_2.html#K16" target="_self">MSAlg_meet</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>

<br/>.= 
the <a href="lattices.html#U1">L_meet</a> of <a href="lattices.html#G3">LattStr</a>(# <span class="p1">(<span class="default"><a href="msualg_2.html#K14" target="_self">MSSub</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_2.html#K15" target="_self">MSAlg_join</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_2.html#K16" target="_self">MSAlg_meet</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> #) <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="msualg_2.html#E2:69_1_1_4"><i><font color="Green">E55</font></i></a>, <a class="ref" href="binop_1.html#D2">BINOP_1:def 2</a></i>
<br/>.= 
<font color="Maroon">c<sub>5</sub></font> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">c<sub>4</sub></font>

;<br/>


</div><b>end;</b></div>
<a NAME="E5:69_1_1"/><i><font color="Green">E55</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="msualg_2.html#K14" target="_self">MSSub</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_2.html#K15" target="_self">MSAlg_join</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_2.html#K16" target="_self">MSAlg_meet</a> <font color="Maroon">c<sub>2</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="69_1_1_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</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="msualg_2.html#K14" target="_self">MSSub</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_2.html#K15" target="_self">MSAlg_join</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_2.html#K16" target="_self">MSAlg_meet</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> #);<br/>

<b>reconsider </b><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>, <font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>6</sub></font> as    <a href="subset_1.html#M1">Element</a> of  <a href="msualg_2.html#K14" target="_self">MSSub</a> <font color="Maroon">c<sub>2</sub></font> ;<br/>
<a NAME="E2:69_1_1_5"/><i><font color="Green">E56</font></i>: 
 <a href="msualg_2.html#K16" target="_self">MSAlg_meet</a> <font color="Maroon">c<sub>2</sub></font> is <a href="binop_1.html#V2">associative</a>
 
<b>by </b><i><a class="ref" href="msualg_2.html#T33" target="_self">Th33</a></i>;<br/>
<b>thus </b><font color="Maroon">c<sub>4</sub></font> <a href="lattices.html#K2">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> = 
the <a href="lattices.html#U1">L_meet</a> of <a href="lattices.html#G3">LattStr</a>(# <span class="p1">(<span class="default"><a href="msualg_2.html#K14" target="_self">MSSub</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_2.html#K15" target="_self">MSAlg_join</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_2.html#K16" target="_self">MSAlg_meet</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> #) <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><a href="msualg_2.html#K16" target="_self">MSAlg_meet</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_2.html#K16" target="_self">MSAlg_meet</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font></span>)</span>

<br/>.= 
the <a href="lattices.html#U1">L_meet</a> of <a href="lattices.html#G3">LattStr</a>(# <span class="p1">(<span class="default"><a href="msualg_2.html#K14" target="_self">MSSub</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_2.html#K15" target="_self">MSAlg_join</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_2.html#K16" target="_self">MSAlg_meet</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> #) <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default">the <a href="lattices.html#U1">L_meet</a> of <a href="lattices.html#G3">LattStr</a>(# <span class="p1">(<span class="default"><a href="msualg_2.html#K14" target="_self">MSSub</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_2.html#K15" target="_self">MSAlg_join</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_2.html#K16" target="_self">MSAlg_meet</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> #) <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="txt" href="msualg_2.html#E2:69_1_1_5"><i><font color="Green">E56</font></i></a>, <a class="ref" href="binop_1.html#D3">BINOP_1:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default">the <a href="lattices.html#U1">L_meet</a> of <a href="lattices.html#G3">LattStr</a>(# <span class="p1">(<span class="default"><a href="msualg_2.html#K14" target="_self">MSSub</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_2.html#K15" target="_self">MSAlg_join</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_2.html#K16" target="_self">MSAlg_meet</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> #) <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">c<sub>6</sub></font>

<br/>.= 
<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> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">c<sub>6</sub></font>

;<br/>


</div><b>end;</b></div>
<a NAME="E6:69_1_1"/>
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="msualg_2.html#K14" target="_self">MSSub</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_2.html#K15" target="_self">MSAlg_join</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_2.html#K16" target="_self">MSAlg_meet</a> <font color="Maroon">c<sub>2</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="69_1_1_6"><b>proof </b></a><div class="add">

<b>let </b><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="msualg_2.html#K14" target="_self">MSSub</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_2.html#K15" target="_self">MSAlg_join</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_2.html#K16" target="_self">MSAlg_meet</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> #);<br/>

<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>5</sub></font> as    <a href="subset_1.html#M1">Element</a> of  <a href="msualg_2.html#K14" target="_self">MSSub</a> <font color="Maroon">c<sub>2</sub></font> ;<br/>
<a NAME="E2:69_1_1_6"/><i><font color="Green">E56</font></i>: 
<span class="p1">(<span class="default"><a href="msualg_2.html#K16" target="_self">MSAlg_meet</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_2.html#K15" target="_self">MSAlg_join</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_1_6_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>7</sub></font> as  <a href="msualg_1.html#V4">strict</a>  <a href="msualg_2.html#M1" target="_self">MSSubAlgebra</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="ref" href="msualg_2.html#D20" target="_self">Def20</a></i>;<br/>
<a NAME="E2:69_1_1_6_1"/>
<span class="p1">(<span class="default"><a href="msualg_2.html#K15" target="_self">MSAlg_join</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="msualg_2.html#K13" target="_self">"\/"</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="ref" href="msualg_2.html#D21" target="_self">Def21</a></i>;<br/>
<b>hence </b><span class="p1">(<span class="default"><a href="msualg_2.html#K16" target="_self">MSAlg_meet</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_2.html#K15" target="_self">MSAlg_join</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> = 
<font color="Maroon">c<sub>8</sub></font> <a href="msualg_2.html#K11" target="_self">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="msualg_2.html#K13" target="_self">"\/"</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="msualg_2.html#D22" target="_self">Def22</a></i>
<br/>.= 
<font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="ref" href="msualg_2.html#T28" target="_self">Th28</a></i>
;<br/><br/>


</div><b>end;</b></div>
<b>thus </b><font color="Maroon">c<sub>4</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#K1">"\/"</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> = 
the <a href="lattices.html#U1">L_meet</a> of <a href="lattices.html#G3">LattStr</a>(# <span class="p1">(<span class="default"><a href="msualg_2.html#K14" target="_self">MSSub</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_2.html#K15" target="_self">MSAlg_join</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_2.html#K16" target="_self">MSAlg_meet</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> #) <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>4</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>

<br/>.= 
<font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="msualg_2.html#E2:69_1_1_6"><i><font color="Green">E56</font></i></a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E7:69_1_1"/><b>then </b>
(  <a href="lattices.html#G3">LattStr</a>(# <span class="p1">(<span class="default"><a href="msualg_2.html#K14" target="_self">MSSub</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_2.html#K15" target="_self">MSAlg_join</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_2.html#K16" target="_self">MSAlg_meet</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> #) is <a href="lattices.html#V3">strict</a> &amp;  <a href="lattices.html#G3">LattStr</a>(# <span class="p1">(<span class="default"><a href="msualg_2.html#K14" target="_self">MSSub</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_2.html#K15" target="_self">MSAlg_join</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_2.html#K16" target="_self">MSAlg_meet</a> <font color="Maroon">c<sub>2</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="msualg_2.html#K14" target="_self">MSSub</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_2.html#K15" target="_self">MSAlg_join</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_2.html#K16" target="_self">MSAlg_meet</a> <font color="Maroon">c<sub>2</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="msualg_2.html#K14" target="_self">MSSub</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_2.html#K15" target="_self">MSAlg_join</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_2.html#K16" target="_self">MSAlg_meet</a> <font color="Maroon">c<sub>2</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="msualg_2.html#K14" target="_self">MSSub</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_2.html#K15" target="_self">MSAlg_join</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_2.html#K16" target="_self">MSAlg_meet</a> <font color="Maroon">c<sub>2</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="msualg_2.html#K14" target="_self">MSSub</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_2.html#K15" target="_self">MSAlg_join</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_2.html#K16" target="_self">MSAlg_meet</a> <font color="Maroon">c<sub>2</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="msualg_2.html#K14" target="_self">MSSub</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_2.html#K15" target="_self">MSAlg_join</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_2.html#K16" target="_self">MSAlg_meet</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> #) is <a href="lattices.html#V9">join-absorbing</a> )
 
<b>by </b><i><a class="txt" href="msualg_2.html#E1:69_1_1"><i><font color="Green">E51</font></i></a>, <a class="txt" href="msualg_2.html#E2:69_1_1"><i><font color="Green">E52</font></i></a>, <a class="txt" href="msualg_2.html#E3:69_1_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="msualg_2.html#E4:69_1_1"><i><font color="Green">E54</font></i></a>, <a class="txt" href="msualg_2.html#E5:69_1_1"><i><font color="Green">E55</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="E8:69_1_1"/>
 <a href="lattices.html#G3">LattStr</a>(# <span class="p1">(<span class="default"><a href="msualg_2.html#K14" target="_self">MSSub</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_2.html#K15" target="_self">MSAlg_join</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_2.html#K16" target="_self">MSAlg_meet</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> #) is  <a href="lattices.html#V3">strict</a> <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>
