
theorem
  for L be Boolean LATTICE for x,a be Element of L st a is atom holds a
  <= x iff not a <= 'not' x
proof
  let L be Boolean LATTICE;
  let x,a be Element of L;
  assume
A1: a is atom;
  thus a <= x implies not a <= 'not' x
  proof
    assume that
A2: a <= x and
A3: a <= 'not' x;
    a = a "\/" Bottom L by WAYBEL_1:3
      .= a "\/" (x "/\" 'not' x) by YELLOW_5:34
      .= (a "\/" x) "/\" ('not' x "\/" a) by WAYBEL_1:5
      .= x "/\" ('not' x "\/" a) by A2,YELLOW_0:24
      .= x "/\" 'not' x by A3,YELLOW_0:24
      .= Bottom L by YELLOW_5:34;
    hence contradiction by A1;
  end;
  thus not a <= 'not' x implies a <= x
  proof
    a <= Top L by YELLOW_0:45;
    then
A4: a <= (x "\/" 'not' x) by YELLOW_5:34;
    assume ( not a <= 'not' x)& not a <= x;
    hence contradiction by A1,A4,WAYBEL14:14;
  end;
end;
