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

<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><a NAME="E1:47_1_1"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">L</font> is <a href="realset2.html#V3">trivial</a>
 ;<br/>

<b>thus </b><a NAME="E2:47_1_1"/>
<font color="Maroon">L</font> is <a href="lattices.html#V9">join-absorbing</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Maroon">y</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>thus </b><a NAME="E1:47_1_1_1"/>
<font color="Maroon">x</font> <a href="lattices.html#K2">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>by </b><i>, <a class="ref" href="realset2.html#D7">REALSET2:def 7</a></i>;<br/>


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

<b>thus </b><a NAME="E3:47_1_1"/>
<font color="Maroon">L</font> is <a href="lattices.html#V8">meet-absorbing</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Maroon">y</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#D8">LATTICES:def 8</a><br/>

<b>thus </b><a NAME="E1:47_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">y</font></span>)</span> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 <b>by </b><i>, <a class="ref" href="realset2.html#D7">REALSET2:def 7</a></i>;<br/>


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

<a NAME="E4:47_1_1"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">L</font> is <a href="lattices.html#V15">bounded</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_1_3"><b>proof </b></a><div class="add">

<a NAME="E1:47_1_1_3"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">L</font> is <a href="lattices.html#V13">lower-bounded</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_1_3_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">c</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/>
<b>take </b>
<font color="Maroon">c</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="lattices.html#D13">LATTICES:def 13</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>;<br/>

<b>thus </b><a NAME="E2:47_1_1_3_1"/>
( <font color="Maroon">c</font> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> &amp; <font color="Maroon">a</font> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> )
 <b>by </b><i>, <a class="ref" href="realset2.html#D7">REALSET2:def 7</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:47_1_1_3"/>
<font color="Maroon">L</font> is <a href="lattices.html#V14">upper-bounded</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_1_3_2"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">c</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/>
<b>take </b>
<font color="Maroon">c</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="lattices.html#D14">LATTICES:def 14</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>;<br/>

<b>thus </b><a NAME="E2:47_1_1_3_2"/>
( <font color="Maroon">c</font> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> &amp; <font color="Maroon">a</font> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> )
 <b>by </b><i>, <a class="ref" href="realset2.html#D7">REALSET2:def 7</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:47_1_1_3"/>
<font color="Maroon">L</font> is <a href="lattices.html#V15">bounded</a>
 <b>by </b><i>, <a class="ref" href="lattices.html#D15">LATTICES:def 15</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:47_1_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">L</font> is <a href="lattices.html#V16">complemented</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">b</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#D19">LATTICES:def 19</a><br/>

<b>consider </b><font color="Maroon">a</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/>
<b>take </b>
<font color="Maroon">a</font>
;<br/>

<a NAME="E2:47_1_1_4"/>
( <font color="Maroon">a</font> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="lattices.html#K6">Top</a> <font color="Maroon">L</font> &amp; <font color="Maroon">b</font> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="lattices.html#K6">Top</a> <font color="Maroon">L</font> &amp; <font color="Maroon">a</font> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="lattices.html#K5">Bottom</a> <font color="Maroon">L</font> &amp; <font color="Maroon">b</font> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="lattices.html#K5">Bottom</a> <font color="Maroon">L</font> )
 
<b>by </b><i>, <a class="ref" href="realset2.html#D7">REALSET2:def 7</a></i>;<br/>
<b>hence </b><a NAME="E3:47_1_1_4"/>
<font color="Maroon">a</font> <a href="lattices.html#R2">is_a_complement_of</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="lattices.html#D18">LATTICES:def 18</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:47_1_1"/>
<font color="Maroon">L</font> is <a href="lattices.html#V11">distributive</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_1_5"><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>, <font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Maroon">c</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#D11">LATTICES:def 11</a><br/>

<b>thus </b><a NAME="E1:47_1_1_5"/>
<font color="Maroon">a</font> <a href="lattices.html#K2">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">c</font></span>)</span> <a href="hidden.html#R1">=</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> <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">c</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="realset2.html#D7">REALSET2:def 7</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:47_1_1"/>
<font color="Maroon">L</font> is <a href="lattices.html#V17">Boolean</a>
 <b>by </b><i>, , <a class="ref" href="lattices.html#D20">LATTICES:def 20</a></i>;<br/>


</div>
