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<b>let </b><font color="Maroon" title="c1">L</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="lattices.html#L3" title="LATTICES:struct.3">LattStr</a> ;<br/>

<b>assume </b><a NAME="E1:47_1_1"/><i><font color="Green" title="E38">A1</font></i>: 
<font color="Maroon" title="c1">L</font> is <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a>
 ;<br/>

<b>thus </b><a NAME="E2:47_1_1"/>
<font color="Maroon" title="c1">L</font> is <a href="lattices.html#V9" title="LATTICES:attr.9">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" title="c2">x</font>, <font color="Maroon" title="c3">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattices.html#D9" title="LATTICES:def.9">LATTICES:def 9</a><br/>

<b>thus </b><a NAME="E1:47_1_1_1"/>
<font color="Maroon" title="c2">x</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font>
 <b>by </b><i><a class="txt" href="sheffer1.html#E1:47_1_1"><i><font color="Green" title="E38">A1</font></i></a>, <a class="ref" href="struct_0.html#D10" title="STRUCT_0:def.10">STRUCT_0:def 10</a></i>;<br/>


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

<b>thus </b><a NAME="E3:47_1_1"/>
<font color="Maroon" title="c1">L</font> is <a href="lattices.html#V8" title="LATTICES:attr.8">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" title="c2">x</font>, <font color="Maroon" title="c3">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattices.html#D8" title="LATTICES:def.8">LATTICES:def 8</a><br/>

<b>thus </b><a NAME="E1:47_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">y</font>
 <b>by </b><i><a class="txt" href="sheffer1.html#E1:47_1_1"><i><font color="Green" title="E38">A1</font></i></a>, <a class="ref" href="struct_0.html#D10" title="STRUCT_0:def.10">STRUCT_0:def 10</a></i>;<br/>


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

<a NAME="E4:47_1_1"/><i><font color="Green" title="E39">A2</font></i>: 
<font color="Maroon" title="c1">L</font> is <a href="lattices.html#V15" title="LATTICES:attr.15">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" title="E40">A3</font></i>: 
<font color="Maroon" title="c1">L</font> is <a href="lattices.html#V13" title="LATTICES:attr.13">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" title="c2">c</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>;<br/>
<b>take </b>
<font color="Maroon" title="c2">c</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="lattices.html#D13" title="LATTICES:def.13">LATTICES:def 13</a><br/>

<b>let </b><font color="Maroon" title="c3">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>;<br/>

<b>thus </b><a NAME="E2:47_1_1_3_1"/>
( <font color="Maroon" title="c2">c</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">c</font> &amp; <font color="Maroon" title="c3">a</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c2">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">c</font> )
 <b>by </b><i><a class="txt" href="sheffer1.html#E1:47_1_1"><i><font color="Green" title="E38">A1</font></i></a>, <a class="ref" href="struct_0.html#D10" title="STRUCT_0:def.10">STRUCT_0:def 10</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:47_1_1_3"/>
<font color="Maroon" title="c1">L</font> is <a href="lattices.html#V14" title="LATTICES:attr.14">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" title="c2">c</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>;<br/>
<b>take </b>
<font color="Maroon" title="c2">c</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="lattices.html#D14" title="LATTICES:def.14">LATTICES:def 14</a><br/>

<b>let </b><font color="Maroon" title="c3">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>;<br/>

<b>thus </b><a NAME="E2:47_1_1_3_2"/>
( <font color="Maroon" title="c2">c</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">c</font> &amp; <font color="Maroon" title="c3">a</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c2">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">c</font> )
 <b>by </b><i><a class="txt" href="sheffer1.html#E1:47_1_1"><i><font color="Green" title="E38">A1</font></i></a>, <a class="ref" href="struct_0.html#D10" title="STRUCT_0:def.10">STRUCT_0:def 10</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:47_1_1_3"/>
<font color="Maroon" title="c1">L</font> is <a href="lattices.html#V15" title="LATTICES:attr.15">bounded</a>
 <b>by </b><i><a class="txt" href="sheffer1.html#E1:47_1_1_3"><i><font color="Green" title="E40">A3</font></i></a>, <a class="ref" href="lattices.html#D15" title="LATTICES:def.15">LATTICES:def 15</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:47_1_1"/><i><font color="Green" title="E40">A4</font></i>: 
<font color="Maroon" title="c1">L</font> is <a href="lattices.html#V16" title="LATTICES:attr.16">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" title="c2">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattices.html#D19" title="LATTICES:def.19">LATTICES:def 19</a><br/>

<b>consider </b><font color="Maroon" title="c3">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>;<br/>
<b>take </b>
<font color="Maroon" title="c3">a</font>
;<br/>

<a NAME="E2:47_1_1_4"/>
( <font color="Maroon" title="c3">a</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <font color="Maroon" title="c1">L</font> &amp; <font color="Maroon" title="c2">b</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <font color="Maroon" title="c1">L</font> &amp; <font color="Maroon" title="c3">a</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c1">L</font> &amp; <font color="Maroon" title="c2">b</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c1">L</font> )
 
<b>by </b><i><a class="txt" href="sheffer1.html#E1:47_1_1"><i><font color="Green" title="E38">A1</font></i></a>, <a class="ref" href="struct_0.html#D10" title="STRUCT_0:def.10">STRUCT_0:def 10</a></i>;<br/>
<b>hence </b><a NAME="E3:47_1_1_4"/>
<font color="Maroon" title="c3">a</font> <a href="lattices.html#R2" title="LATTICES:pred.2">is_a_complement_of</a> <font color="Maroon" title="c2">b</font>
 <b>by </b><i><a class="ref" href="lattices.html#D18" title="LATTICES:def.18">LATTICES:def 18</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:47_1_1"/>
<font color="Maroon" title="c1">L</font> is <a href="lattices.html#V11" title="LATTICES:attr.11">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" title="c2">a</font>, <font color="Maroon" title="c3">b</font>, <font color="Maroon" title="c4">c</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattices.html#D11" title="LATTICES:def.11">LATTICES:def 11</a><br/>

<b>thus </b><a NAME="E1:47_1_1_5"/>
<font color="Maroon" title="c2">a</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c4">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c4">c</font></span>)</span>
 <b>by </b><i><a class="txt" href="sheffer1.html#E1:47_1_1"><i><font color="Green" title="E38">A1</font></i></a>, <a class="ref" href="struct_0.html#D10" title="STRUCT_0:def.10">STRUCT_0:def 10</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:47_1_1"/>
<font color="Maroon" title="c1">L</font> is <a href="lattices.html#V17" title="LATTICES:attr.17">Boolean</a>
 <b>by </b><i><a class="txt" href="sheffer1.html#E4:47_1_1"><i><font color="Green" title="E39">A2</font></i></a>, <a class="txt" href="sheffer1.html#E5:47_1_1"><i><font color="Green" title="E40">A4</font></i></a>, <a class="ref" href="lattices.html#D20" title="LATTICES:def.20">LATTICES:def 20</a></i>;<br/>


</div>
