
theorem Th37:
  for S being Scott complete TopLattice for T being Lawson correct
TopAugmentation of S for A being Subset of S st A is open for C being Subset of
  T st C = A holds C is open
proof
  let S be Scott complete TopLattice;
  let T be Lawson correct TopAugmentation of S;
  let A be Subset of S;
  assume
A1: A in the topology of S;
  let C be Subset of T;
  assume
A2: C = A;
  (sigma T) \/ omega T is prebasis of T by Def3;
  then
A3: (sigma T) \/ omega T c= the topology of T by TOPS_2:64;
  the RelStr of S = the RelStr of T by YELLOW_9:def 4;
  then S is Scott TopAugmentation of T by YELLOW_9:def 4;
  then A in sigma T by A1,YELLOW_9:51;
  then C in (sigma T) \/ omega T by A2,XBOOLE_0:def 3;
  hence C in the topology of T by A3;
end;
