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<b>let </b><font color="Maroon">L</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">E17</font></i>: 
the <a href="lattices.html#U2">L_join</a> of <font color="Maroon">L</font> is <a href="binop_1.html#V1">commutative</a>
 <b>and </b><br/><a NAME="E2:14"/><i><font color="Green">E18</font></i>: 
the <a href="lattices.html#U2">L_join</a> of <font color="Maroon">L</font> is <a href="binop_1.html#V2">associative</a>
 <b>and </b><br/><a NAME="E3:14"/><i><font color="Green">E19</font></i>: 
the <a href="lattices.html#U1">L_meet</a> of <font color="Maroon">L</font> is <a href="binop_1.html#V1">commutative</a>
 <b>and </b><br/><a NAME="E4:14"/><i><font color="Green">E22</font></i>: 
the <a href="lattices.html#U1">L_meet</a> of <font color="Maroon">L</font> is <a href="binop_1.html#V2">associative</a>
 <b>and </b><br/><a NAME="E5:14"/><i><font color="Green">E23</font></i>: 
the <a href="lattices.html#U2">L_join</a> of <font color="Maroon">L</font> <a href="lattice2.html#R1" target="_self">absorbs</a> the <a href="lattices.html#U1">L_meet</a> of <font color="Maroon">L</font>
 <b>and </b><br/><a NAME="E6:14"/><i><font color="Green">E42</font></i>: 
the <a href="lattices.html#U1">L_meet</a> of <font color="Maroon">L</font> <a href="lattice2.html#R1" target="_self">absorbs</a> the <a href="lattices.html#U2">L_join</a> of <font color="Maroon">L</font>
 ;<br/>

<b>thus </b><a NAME="E7:14"/><i><font color="Green">E43</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> holds  <font color="Olive">a</font> <a href="lattices.html#K1">"\/"</a> <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> <a href="lattices.html#K1">"\/"</a> <font color="Olive">a</font>
 <b>by </b><i>, <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">E44</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> holds  <font color="Olive">a</font> <a href="lattices.html#K2">"/\"</a> <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> <a href="lattices.html#K2">"/\"</a> <font color="Olive">a</font>
 
<b>by </b><i><a class="ref" href="lattice2.html#T2" target="_self">Th2</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">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> holds  <font color="Olive">a</font> <a href="lattices.html#K1">"\/"</a> <span class="p1">(<span class="default"><font color="Olive">b</font> <a href="lattices.html#K1">"\/"</a> <font color="Olive">c</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">a</font> <a href="lattices.html#K1">"\/"</a> <font color="Olive">b</font></span>)</span> <a href="lattices.html#K1">"\/"</a> <font color="Olive">c</font>
 <b>by </b><i>, <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">a</font>, <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> holds  <span class="p1">(<span class="default"><font color="Olive">a</font> <a href="lattices.html#K2">"/\"</a> <font color="Olive">b</font></span>)</span> <a href="lattices.html#K1">"\/"</a> <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</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">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/><b>let </b><font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/>



<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">b</font></span>)</span> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">b</font> = 
<font color="Maroon">b</font> <a href="lattices.html#K1">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">b</font></span>)</span>
<b>by </b><i><a class="txt" href="lattice2.html#E3:14"><i><font color="Green">E19</font></i></a></i>
<br/>.= 
<font color="Maroon">b</font> <a href="lattices.html#K1">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">a</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">b</font>
<b>by </b><i><a class="txt" href="lattice2.html#E1:14"><i><font color="Green">E17</font></i></a>, </i>
;<br/>


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

<b>thus </b><a NAME="E11:14"/>
for <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> holds  <font color="Olive">a</font> <a href="lattices.html#K2">"/\"</a> <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> <a href="lattices.html#K2">"/\"</a> <font color="Olive">a</font>
 <b>by </b><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">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> holds  <font color="Olive">a</font> <a href="lattices.html#K2">"/\"</a> <span class="p1">(<span class="default"><font color="Olive">b</font> <a href="lattices.html#K2">"/\"</a> <font color="Olive">c</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">a</font> <a href="lattices.html#K2">"/\"</a> <font color="Olive">b</font></span>)</span> <a href="lattices.html#K2">"/\"</a> <font color="Olive">c</font>
 <b>by </b><i><a class="ref" href="lattice2.html#T5" target="_self">Th5</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">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</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">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/>



<b>thus </b><a NAME="E13:14"/>
<font color="Maroon">a</font> <a href="lattices.html#K2">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="txt" href="lattice2.html#E2:14"><i><font color="Green">E18</font></i></a>, </i>;<br/>


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