
theorem
  for S being up-complete /\-complete Semilattice,
  T being Scott TopAugmentation of S holds sigma S = the topology of T
proof
  let S be up-complete /\-complete Semilattice;
  let T be Scott TopAugmentation of S;
  thus sigma S c= the topology of T
  proof
    let e be object;
    assume
A1: e in sigma S;
    then reconsider A = e as Subset of S;
A2: the RelStr of S = the RelStr of T by YELLOW_9:def 4;
    then reconsider A9 = A as Subset of T;
    A is inaccessible upper by A1,Th10;
    then A9 is inaccessible upper by A2,WAYBEL_0:25,YELLOW_9:47;
    hence thesis by PRE_TOPC:def 2;
  end;
  let e be object;
  assume
A3: e in the topology of T;
  then reconsider A = e as Subset of T;
A4: A is open by A3,PRE_TOPC:def 2;
A5: the RelStr of S = the RelStr of T by YELLOW_9:def 4;
  then reconsider A9 = A as Subset of S;
  A9 is inaccessible
  proof
    let D be non empty directed Subset of S such that
A6: sup D in A9;
    reconsider E = D as Subset of T by A5;
    ex a being Element of S st a is_>=_than D &
    for b being Element of S st b is_>=_than D holds a <= b
    by WAYBEL_0:def 39;
    then
A7: ex_sup_of D,S by YELLOW_0:15;
A8: E is directed by A5,WAYBEL_0:3;
    sup D = sup E by A5,A7,YELLOW_0:26;
    hence thesis by A4,A6,A8,WAYBEL11:def 1;
  end;
  then A9 is inaccessible upper by A4,A5,WAYBEL_0:25;
  hence thesis by Th10;
end;
