
theorem Th19:
  for R being complete connected LATTICE,
  T being Scott TopAugmentation of R, x being Element of T holds
  (downarrow x)` is open
proof
  let R be complete connected LATTICE,
  T be Scott TopAugmentation of R, x be Element of T;
  reconsider S = downarrow x as directly_closed lower Subset of T by WAYBEL11:8
  ;
  S` is open;
  hence thesis;
end;
