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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="struct_0.html#V3">empty</a>  <a href="lattices.html#L3">LattStr</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:14"/><i><font color="Green">E13</font></i>: 
the <a href="lattices.html#U2">L_join</a> of <font color="Maroon">c<sub>1</sub></font> is <a href="binop_1.html#V1">commutative</a>
 <b>and </b><br/><a NAME="E2:14"/><i><font color="Green">E14</font></i>: 
the <a href="lattices.html#U2">L_join</a> of <font color="Maroon">c<sub>1</sub></font> is <a href="binop_1.html#V2">associative</a>
 <b>and </b><br/><a NAME="E3:14"/><i><font color="Green">E15</font></i>: 
the <a href="lattices.html#U1">L_meet</a> of <font color="Maroon">c<sub>1</sub></font> is <a href="binop_1.html#V1">commutative</a>
 <b>and </b><br/><a NAME="E4:14"/><i><font color="Green">E16</font></i>: 
the <a href="lattices.html#U1">L_meet</a> of <font color="Maroon">c<sub>1</sub></font> is <a href="binop_1.html#V2">associative</a>
 <b>and </b><br/><a NAME="E5:14"/><i><font color="Green">E17</font></i>: 
the <a href="lattices.html#U2">L_join</a> of <font color="Maroon">c<sub>1</sub></font> <a href="lattice2.html#R1" target="_self">absorbs</a> the <a href="lattices.html#U1">L_meet</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E6:14"/><i><font color="Green">E18</font></i>: 
the <a href="lattices.html#U1">L_meet</a> of <font color="Maroon">c<sub>1</sub></font> <a href="lattice2.html#R1" target="_self">absorbs</a> the <a href="lattices.html#U2">L_join</a> of <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<b>thus </b><a NAME="E7:14"/><i><font color="Green">E19</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 <font color="Maroon">c<sub>1</sub></font> 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>
 <b>by </b><i><a class="txt" href="lattice2.html#E1:14"><i><font color="Green">E13</font></i></a>, <a class="ref" href="binop_1.html#D2">BINOP_1:def 2</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattices.html#D4">LATTICES:def 4</a>,<a class="ref" href="lattices.html#D10">LATTICES:def 10</a><br/>

<a NAME="E8:14"/><i><font color="Green">E20</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 <font color="Maroon">c<sub>1</sub></font> 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>
 
<b>by </b><i><a class="txt" href="lattice2.html#E3:14"><i><font color="Green">E15</font></i></a>, <a class="ref" href="binop_1.html#D2">BINOP_1:def 2</a></i>;<br/>
<b>thus </b><a NAME="E9:14"/>
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 <font color="Maroon">c<sub>1</sub></font> 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>
 <b>by </b><i><a class="txt" href="lattice2.html#E2:14"><i><font color="Green">E14</font></i></a>, <a class="ref" href="binop_1.html#D3">BINOP_1:def 3</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattices.html#D5">LATTICES:def 5</a><br/>

<b>thus </b><a NAME="E10:14"/>
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 <font color="Maroon">c<sub>1</sub></font> 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>
  <i><font color="Red">:: according to </font></i><a class="ref" href="lattices.html#D8">LATTICES:def 8</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1"><b>proof </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 <font color="Maroon">c<sub>1</sub></font>;<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> = 
<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>2</sub></font> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="lattice2.html#E7:14"><i><font color="Green">E19</font></i></a></i>
<br/>.= 
<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>3</sub></font> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="lattice2.html#E8:14"><i><font color="Green">E20</font></i></a></i>
<br/>.= 
the <a href="lattices.html#U2">L_join</a> of <font color="Maroon">c<sub>1</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default">the <a href="lattices.html#U1">L_meet</a> of <font color="Maroon">c<sub>1</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>

<br/>.= 
<font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="lattice2.html#E5:14"><i><font color="Green">E17</font></i></a>, <a class="ref" href="lattice2.html#D1" target="_self">Def1</a></i>
;<br/>


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

<b>thus </b><a NAME="E11:14"/>
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 <font color="Maroon">c<sub>1</sub></font> 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>
 <b>by </b><i><a class="txt" href="lattice2.html#E8:14"><i><font color="Green">E20</font></i></a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattices.html#D6">LATTICES:def 6</a><br/>

<b>thus </b><a NAME="E12:14"/>
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 <font color="Maroon">c<sub>1</sub></font> 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>
 <b>by </b><i><a class="txt" href="lattice2.html#E4:14"><i><font color="Green">E16</font></i></a>, <a class="ref" href="binop_1.html#D3">BINOP_1:def 3</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattices.html#D7">LATTICES:def 7</a><br/>

<b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattices.html#D9">LATTICES:def 9</a><br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<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> = 
the <a href="lattices.html#U1">L_meet</a> of <font color="Maroon">c<sub>1</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default">the <a href="lattices.html#U2">L_join</a> of <font color="Maroon">c<sub>1</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>

<br/>.= 
<font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="lattice2.html#E6:14"><i><font color="Green">E18</font></i></a>, <a class="ref" href="lattice2.html#D1" target="_self">Def1</a></i>
;<br/>


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