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

<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#V10" title="LATTICES:attr.10">Lattice-like</a> <a href="lattices.html#V17" title="LATTICES:attr.17">Boolean</a> <a href="sheffer1.html#V1" title="SHEFFER1:attr.1">upper-bounded'</a> <a href="sheffer1.html#V2" title="SHEFFER1:attr.2">lower-bounded'</a> <a href="sheffer1.html#V3" title="SHEFFER1:attr.3">distributive'</a>  <a href="lattices.html#L3" title="LATTICES:struct.3">LattStr</a> ;<br/>

<a NAME="E1:34"/>
for <font color="Olive" title="b1">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>  ex <font color="Olive" title="b2">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <font color="Olive" title="b2">a</font> <a href="sheffer1.html#R1" title="SHEFFER1:pred.1">is_a_complement'_of</a> <font color="Olive" title="b1">b</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1"><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>;<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><b> such that </b><br/><a NAME="E2:34_1"/><i><font color="Green" title="E33">A1</font></i>: 
<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#D19" title="LATTICES:def.19">LATTICES:def 19</a></i>;<br/>
<a NAME="E3:34_1"/>
( <font color="Maroon" title="c2">b</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</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="c2">b</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</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#E2:34_1"><i><font color="Green" title="E33">A1</font></i></a>, <a class="ref" href="lattices.html#D18" title="LATTICES:def.18">LATTICES:def 18</a></i>;<br/>
<a NAME="E4:34_1"/><b>then </b>
( <font color="Maroon" title="c2">b</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="sheffer1.html#K1" title="SHEFFER1:func.1">Top'</a> <font color="Maroon" title="c1">L</font> &amp; <font color="Maroon" title="c2">b</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="sheffer1.html#K2" title="SHEFFER1:func.2">Bot'</a> <font color="Maroon" title="c1">L</font> )
 
<b>by </b><i><a class="ref" href="sheffer1.html#T21" target="_self" title="SHEFFER1:th.21">Th21</a>, <a class="ref" href="sheffer1.html#T22" target="_self" title="SHEFFER1:th.22">Th22</a></i>;<br/>
<a NAME="E5:34_1"/><b>then </b>
<font color="Maroon" title="c3">a</font> <a href="sheffer1.html#R1" title="SHEFFER1:pred.1">is_a_complement'_of</a> <font color="Maroon" title="c2">b</font>
 
<b>by </b><i><a class="ref" href="sheffer1.html#D6" target="_self" title="SHEFFER1:def.6">Def6</a></i>;<br/>
<b>hence </b><a NAME="E6:34_1"/>
 ex <font color="Olive" title="b1">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <font color="Olive" title="b1">a</font> <a href="sheffer1.html#R1" title="SHEFFER1:pred.1">is_a_complement'_of</a> <font color="Maroon" title="c2">b</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:34"/>
<font color="Maroon" title="c1">L</font> is <a href="sheffer1.html#V4" title="SHEFFER1:attr.4">complemented'</a>
 <b>by </b><i><a class="ref" href="sheffer1.html#D7" target="_self" title="SHEFFER1:def.7">Def7</a></i>;<br/>


</div>
