reserve X for set,
  x,y,z for Element of BooleLatt X,
  s for set;
reserve y for Element of BooleLatt X;

theorem
  for x being Element of BooleLatt X holds x` = X \ x
proof
  set B = BooleLatt X;
  let x be Element of B;
A1: x`"/\"x = Bottom B by LATTICES:20;
A2: x"\/"x` = Top B by LATTICES:21;
A3: Bottom B = {} by Th3;
A4: Top B = X by Th4;
A5: x` misses x by A1,A3;
A6: X \ x = (x \ x) \/ (x` \ x) by A2,A4,XBOOLE_1:42;
  x \ x = {} by XBOOLE_1:37;
  hence thesis by A5,A6,XBOOLE_1:83;
end;
