<?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#V4">join-commutative</a> <a href="lattices.html#V6">meet-commutative</a> <a href="lattices.html#V11">distributive</a> <a href="robbins1.html#V6">join-idempotent</a> <a href="sheffer1.html#V1" target="_self">upper-bounded'</a> <a href="sheffer1.html#V2" target="_self">lower-bounded'</a> <a href="sheffer1.html#V3" target="_self">distributive'</a> <a href="sheffer1.html#V4" target="_self">complemented'</a>  <a href="lattices.html#L3">LattStr</a> ;<br/><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>;<br/>



<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:32_1"/>
<font color="Maroon">x</font> <a href="sheffer1.html#R1" target="_self">is_a_complement'_of</a> <font color="Maroon">y</font>
 ;<br/><a NAME="E2:32_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="sheffer1.html#K1" target="_self">Top'</a> <font color="Maroon">L</font> &amp; <font color="Maroon">y</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="sheffer1.html#K1" target="_self">Top'</a> <font color="Maroon">L</font> &amp; <font color="Maroon">x</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="sheffer1.html#K2" target="_self">Bot'</a> <font color="Maroon">L</font> &amp; <font color="Maroon">y</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="sheffer1.html#K2" target="_self">Bot'</a> <font color="Maroon">L</font> )
 <b>by </b><i/>;<br/><a NAME="E3:32_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="lattices.html#K6">Top</a> <font color="Maroon">L</font> &amp; <font color="Maroon">y</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="lattices.html#K6">Top</a> <font color="Maroon">L</font> &amp; <font color="Maroon">x</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="lattices.html#K5">Bottom</a> <font color="Maroon">L</font> &amp; <font color="Maroon">y</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">x</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="sheffer1.html#D3" target="_self">Def3</a></i>;<br/><b>hence </b><a NAME="E4:32_1"/>
<font color="Maroon">x</font> <a href="lattices.html#R2">is_a_complement_of</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="lattices.html#D18">LATTICES:def 18</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:32"/>
<font color="Maroon">x</font> <a href="lattices.html#R2">is_a_complement_of</a> <font color="Maroon">y</font>
 ;<br/>

<a NAME="E3:32"/><b>then </b>
( <font color="Maroon">x</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="lattices.html#K6">Top</a> <font color="Maroon">L</font> &amp; <font color="Maroon">y</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="lattices.html#K6">Top</a> <font color="Maroon">L</font> &amp; <font color="Maroon">x</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="lattices.html#K5">Bottom</a> <font color="Maroon">L</font> &amp; <font color="Maroon">y</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">x</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="lattices.html#D18">LATTICES:def 18</a></i>;<br/>
<a NAME="E4:32"/><b>then </b>
( <font color="Maroon">x</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="sheffer1.html#K1" target="_self">Top'</a> <font color="Maroon">L</font> &amp; <font color="Maroon">y</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="sheffer1.html#K1" target="_self">Top'</a> <font color="Maroon">L</font> &amp; <font color="Maroon">x</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="sheffer1.html#K2" target="_self">Bot'</a> <font color="Maroon">L</font> &amp; <font color="Maroon">y</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="sheffer1.html#K2" target="_self">Bot'</a> <font color="Maroon">L</font> )
 
<b>by </b><i>, <a class="ref" href="sheffer1.html#D3" target="_self">Def3</a></i>;<br/>
<b>hence </b><a NAME="E5:32"/>
<font color="Maroon">x</font> <a href="sheffer1.html#R1" target="_self">is_a_complement'_of</a> <font color="Maroon">y</font>
 <b>by </b><i/>;<br/>


</div>
