
theorem
  for N being Lawson complete TopLattice for S being Scott
TopAugmentation of N for A being Subset of N, J being Subset of S st A = J & J
  is closed holds A is closed
proof
  let N be Lawson complete TopLattice, S be Scott TopAugmentation of N, A be
  Subset of N, J be Subset of S such that
A1: A = J;
  assume J is closed;
  then
A2: [#]S \ J is open;
A3: the RelStr of N = the RelStr of S by YELLOW_9:def 4;
  then N is Lawson correct TopAugmentation of S by YELLOW_9:def 4;
  hence [#]N \ A is open by A1,A3,A2,WAYBEL19:37;
end;
