
theorem Th14:
  for S1, S2 being up-complete antisymmetric non empty reflexive
TopRelStr st the TopRelStr of S1 = the TopRelStr of S2 & S1 is Scott holds S2
  is Scott
proof
  let S1, S2 be up-complete antisymmetric non empty reflexive TopRelStr;
  assume
A1: the TopRelStr of S1 = the TopRelStr of S2;
  assume
A2: S1 is Scott;
  let T be Subset of S2;
  reconsider T1=T as Subset of S1 by A1;
A3: the RelStr of S1 = the RelStr of S2 by A1;
  thus T is open implies T is inaccessible upper
  proof
    assume T is open;
    then T in the topology of S2;
    then T1 is open by A1;
    hence thesis by A3,A2,WAYBEL_0:25,YELLOW_9:47;
  end;
  thus T is inaccessible upper implies T is open
  proof
    assume T is inaccessible upper;
    then T1 is inaccessible upper by A3,WAYBEL_0:25,YELLOW_9:47;
    then T1 in the topology of S1 by A2,PRE_TOPC:def 2;
    hence thesis by A1;
  end;
end;
