
theorem Th49:
  for S being complete LATTICE, T being TopAugmentation of S
  st the topology of T = sigma S holds T is Scott
proof
  let R be complete LATTICE;
  let T be TopAugmentation of R such that
A1: the topology of T = sigma R;
A2: the RelStr of T = the RelStr of R by Def4;
  T is Scott
  proof
    let S be Subset of T;
    reconsider A = S as Subset of R by A2;
    thus S is open implies S is inaccessible upper
    by A1,WAYBEL11:31,A2,Th47,WAYBEL_0:25;
    assume S is inaccessible upper;
    then A is inaccessible upper by A2,Th47,WAYBEL_0:25;
    hence S in the topology of T by A1,WAYBEL11:31;
  end;
  hence thesis;
end;
