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<div><i><font color="Green">E4</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K1" target="_self">BooleLatt</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/><b>thus </b><font color="Maroon">c<sub>2</sub></font> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">c<sub>3</sub></font> = 
<font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="ref" href="lattice3.html#T1" target="_self">Th1</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="ref" href="lattice3.html#T1" target="_self">Th1</a></i>
;<br/></div><b>end;</b></div>
<div><i><font color="Green">E5</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1_1_2"><b>now </b></a><div class="add"><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 <span class="p1">(<span class="default"><a href="lattice3.html#K1" target="_self">BooleLatt</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/><b>thus </b><font color="Maroon">c<sub>2</sub></font> <a href="lattices.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>2</sub></font> <a href="xboole_0.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>
<b>by </b><i><a class="ref" href="lattice3.html#T1" target="_self">Th1</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#T1" target="_self">Th1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="ref" href="xboole_1.html#T4">XBOOLE_1:4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="ref" href="lattice3.html#T1" target="_self">Th1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="ref" href="lattice3.html#T1" target="_self">Th1</a></i>
;<br/></div><b>end;</b></div>
<div><i><font color="Green">E6</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K1" target="_self">BooleLatt</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">c<sub>3</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="ref" href="lattice3.html#T1" target="_self">Th1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="ref" href="lattice3.html#T1" target="_self">Th1</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="ref" href="xboole_1.html#T22">XBOOLE_1:22</a></i>
;<br/></div><b>end;</b></div>
<div><i><font color="Green">E7</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1_1_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K1" target="_self">BooleLatt</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/><b>thus </b><font color="Maroon">c<sub>2</sub></font> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">c<sub>3</sub></font> = 
<font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="ref" href="lattice3.html#T1" target="_self">Th1</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="ref" href="lattice3.html#T1" target="_self">Th1</a></i>
;<br/></div><b>end;</b></div>
<div><i><font color="Green">E8</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1_1_5"><b>now </b></a><div class="add"><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 <span class="p1">(<span class="default"><a href="lattice3.html#K1" target="_self">BooleLatt</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/><b>thus </b><font color="Maroon">c<sub>2</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#K2">"/\"</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> = 
<font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#K3">/\</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>
<b>by </b><i><a class="ref" href="lattice3.html#T1" target="_self">Th1</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#T1" target="_self">Th1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="ref" href="xboole_1.html#T16">XBOOLE_1:16</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="ref" href="lattice3.html#T1" target="_self">Th1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="ref" href="lattice3.html#T1" target="_self">Th1</a></i>
;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1_1_6"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K1" target="_self">BooleLatt</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/><b>thus </b><font color="Maroon">c<sub>2</sub></font> <a href="lattices.html#K2">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> = 
<font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#T1" target="_self">Th1</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#T1" target="_self">Th1</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="ref" href="xboole_1.html#T21">XBOOLE_1:21</a></i>
;<br/></div><b>end;</b></div>
<a NAME="E7:5_1_1"/><b>then </b>
(  <a href="lattice3.html#K1" target="_self">BooleLatt</a> <font color="Maroon">c<sub>1</sub></font> is <a href="lattices.html#V4">join-commutative</a> &amp;  <a href="lattice3.html#K1" target="_self">BooleLatt</a> <font color="Maroon">c<sub>1</sub></font> is <a href="lattices.html#V5">join-associative</a> &amp;  <a href="lattice3.html#K1" target="_self">BooleLatt</a> <font color="Maroon">c<sub>1</sub></font> is <a href="lattices.html#V8">meet-absorbing</a> &amp;  <a href="lattice3.html#K1" target="_self">BooleLatt</a> <font color="Maroon">c<sub>1</sub></font> is <a href="lattices.html#V6">meet-commutative</a> &amp;  <a href="lattice3.html#K1" target="_self">BooleLatt</a> <font color="Maroon">c<sub>1</sub></font> is <a href="lattices.html#V7">meet-associative</a> &amp;  <a href="lattice3.html#K1" target="_self">BooleLatt</a> <font color="Maroon">c<sub>1</sub></font> is <a href="lattices.html#V9">join-absorbing</a> )
 
<b>by </b><i><a class="txt" href="lattice3.html#E1:5_1_1"><i><font color="Green">E4</font></i></a>, <a class="txt" href="lattice3.html#E2:5_1_1"><i><font color="Green">E5</font></i></a>, <a class="txt" href="lattice3.html#E3:5_1_1"><i><font color="Green">E6</font></i></a>, <a class="txt" href="lattice3.html#E4:5_1_1"><i><font color="Green">E7</font></i></a>, <a class="txt" href="lattice3.html#E5:5_1_1"><i><font color="Green">E8</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="E8:5_1_1"/>
 <a href="lattice3.html#K1" target="_self">BooleLatt</a> <font color="Maroon">c<sub>1</sub></font> is <a href="lattices.html#V10">Lattice-like</a>
 <b>by </b><i><a class="ref" href="lattices.html#D10">LATTICES:def 10</a></i>;<br/>


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