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<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="lattices.html#NM8">B_Lattice</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <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 <font color="Maroon">c<sub>1</sub></font>;<br/>







<b>thus </b><a NAME="E1:55"/>
(  not <font color="Maroon">c<sub>2</sub></font> <a href="boolealg.html#R3" target="_self">meets</a> <font color="Maroon">c<sub>3</sub></font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c<sub>4</sub></font> or <font color="Maroon">c<sub>2</sub></font> <a href="boolealg.html#R3" target="_self">meets</a> <font color="Maroon">c<sub>3</sub></font> or <font color="Maroon">c<sub>2</sub></font> <a href="boolealg.html#R3" target="_self">meets</a> <font color="Maroon">c<sub>4</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:55_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="boolealg.html#R3" target="_self">meets</a> <font color="Maroon">c<sub>3</sub></font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<a NAME="E2:55_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="lattices.html#K5">Bottom</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="boolealg.html#D4" target="_self">Def4</a></i>;<br/>
<a NAME="E3:55_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="lattices.html#K5">Bottom</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="lattices.html#D11">LATTICES:def 11</a></i>;<br/>
<a NAME="E4:55_1"/><b>then </b>
( <font color="Maroon">c<sub>2</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="lattices.html#K5">Bottom</a> <font color="Maroon">c<sub>1</sub></font> or <font color="Maroon">c<sub>2</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="lattices.html#K5">Bottom</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="ref" href="lattices.html#T39">LATTICES:39</a></i>;<br/>
<b>hence </b><a NAME="E5:55_1"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="boolealg.html#R3" target="_self">meets</a> <font color="Maroon">c<sub>3</sub></font> or <font color="Maroon">c<sub>2</sub></font> <a href="boolealg.html#R3" target="_self">meets</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="ref" href="boolealg.html#D4" target="_self">Def4</a></i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E2:55"/><i><font color="Green">E26</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="boolealg.html#R3" target="_self">meets</a> <font color="Maroon">c<sub>3</sub></font> or <font color="Maroon">c<sub>2</sub></font> <a href="boolealg.html#R3" target="_self">meets</a> <font color="Maroon">c<sub>4</sub></font> )
 ;<br/>

<a NAME="E3:55"/><i><font color="Green">E27</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="boolealg.html#R3" target="_self">meets</a> <font color="Maroon">c<sub>3</sub></font> implies <font color="Maroon">c<sub>2</sub></font> <a href="boolealg.html#R3" target="_self">meets</a> <font color="Maroon">c<sub>3</sub></font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c<sub>4</sub></font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:55_2"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="lattices.html#K5">Bottom</a> <font color="Maroon">c<sub>1</sub></font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="boolealg.html#D4" target="_self">BOOLEALG:def 4</a><br/>

<a NAME="E2:55_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>3</sub></font> <a href="lattices.html#R3">[=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="lattices.html#T22">LATTICES:22</a></i>;<br/>
<a NAME="E3:55_2"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>3</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>2</sub></font> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="lattices.html#D11">LATTICES:def 11</a></i>;<br/>
<a NAME="E4:55_2"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="lattices.html#K5">Bottom</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="boolealg.html#E1:55_2"><i><font color="Green">E28</font></i></a>, <a class="ref" href="boolealg.html#T25" target="_self">Th25</a></i>;<br/>
<b>hence </b><a NAME="E5:55_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="boolealg.html#R3" target="_self">meets</a> <font color="Maroon">c<sub>3</sub></font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="boolealg.html#D4" target="_self">Def4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:55"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="boolealg.html#R3" target="_self">meets</a> <font color="Maroon">c<sub>4</sub></font> implies <font color="Maroon">c<sub>2</sub></font> <a href="boolealg.html#R3" target="_self">meets</a> <font color="Maroon">c<sub>3</sub></font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c<sub>4</sub></font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:55_3"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="lattices.html#K5">Bottom</a> <font color="Maroon">c<sub>1</sub></font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="boolealg.html#D4" target="_self">BOOLEALG:def 4</a><br/>

<a NAME="E2:55_3"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="boolealg.html#R1" target="_self">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="lattices.html#D11">LATTICES:def 11</a></i>;<br/>
<a NAME="E3:55_3"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="lattices.html#K5">Bottom</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="boolealg.html#E1:55_3"><i><font color="Green">E28</font></i></a>, <a class="ref" href="boolealg.html#T27" target="_self">Th27</a></i>;<br/>
<b>hence </b><a NAME="E4:55_3"/>
<font color="Maroon">c<sub>2</sub></font> <a href="boolealg.html#R3" target="_self">meets</a> <font color="Maroon">c<sub>3</sub></font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="boolealg.html#D4" target="_self">Def4</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:55"/>
<font color="Maroon">c<sub>2</sub></font> <a href="boolealg.html#R3" target="_self">meets</a> <font color="Maroon">c<sub>3</sub></font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="boolealg.html#E2:55"><i><font color="Green">E26</font></i></a>, <a class="txt" href="boolealg.html#E3:55"><i><font color="Green">E27</font></i></a></i>;<br/>


</div>
