
theorem Th39:
  for T being Lawson complete TopLattice for x being Element of
  T holds (uparrow x)` is open & (downarrow x)` is open & {x}` is open
proof
  let T be Lawson complete TopLattice;
  let x be Element of T;
A1: downarrow x is closed by Th38;
A2: {x} is closed by Th38;
  uparrow x is closed by Th38;
  hence thesis by A1,A2;
end;
