
theorem Th40:
  for T being Lawson complete continuous TopLattice for x being
  Element of T holds wayabove x is open & (wayabove x)` is closed
proof
  let T be Lawson continuous complete TopLattice;
  let x be Element of T;
  set S = the Scott TopAugmentation of T;
A1: T is TopAugmentation of S by YELLOW_9:45;
A2: the RelStr of S = the RelStr of T by YELLOW_9:def 4;
  then reconsider v = x as Element of S;
  wayabove v is open by WAYBEL11:36;
  hence wayabove x is open by A2,A1,Th37,YELLOW12:13;
  hence thesis;
end;
