
theorem Th11:
  for R being /\-complete up-complete Semilattice,
  T being TopAugmentation of R st the topology of T = sigma R holds T is Scott
proof
  let R be /\-complete up-complete Semilattice;
  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 YELLOW_9:def 4;
  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
    proof
      assume S is open;
      then S in the topology of T by PRE_TOPC:def 2;
      then A is inaccessible upper by A1,Th10;
      hence thesis by A2,WAYBEL_0:25,YELLOW_9:47;
    end;
    assume
A3: S is inaccessible upper;
    A is inaccessible
    proof
      let D be non empty directed Subset of R such that
A4:   sup D in A;
      reconsider E = D as Subset of T by A2;
      ex a being Element of R st a is_>=_than D &
      for b being Element of R st b is_>=_than D holds a <= b
      by WAYBEL_0:def 39;
      then
A5:   ex_sup_of D,R by YELLOW_0:15;
A6:   E is directed by A2,WAYBEL_0:3;
      sup D = sup E by A2,A5,YELLOW_0:26;
      hence thesis by A3,A4,A6;
    end;
    then A is inaccessible upper by A2,A3,WAYBEL_0:25;
    then A in the topology of T by A1,Th10;
    hence thesis by PRE_TOPC:def 2;
  end;
  hence thesis;
end;
