
theorem Th13:
  for L being up-complete non empty Poset for S1, S2 being Scott
  TopAugmentation of L holds the topology of S1 = the topology of S2
proof
  let L be up-complete non empty Poset;
  let S1, S2 be Scott TopAugmentation of L;
A1: the RelStr of S1 = the RelStr of L by YELLOW_9:def 4
    .= the RelStr of S2 by YELLOW_9:def 4;
  thus the topology of S1 c= the topology of S2
  proof
    let x be object;
    assume x in the topology of S1;
    then reconsider y=x as open Subset of S1 by PRE_TOPC:def 2;
    reconsider z=y as Subset of S2 by A1;
    z is inaccessible upper by A1,WAYBEL_0:25,YELLOW_9:47;
    hence thesis by PRE_TOPC:def 2;
  end;
  let x be object;
  assume x in the topology of S2;
  then reconsider y=x as open Subset of S2 by PRE_TOPC:def 2;
  reconsider z=y as Subset of S1 by A1;
  z is inaccessible upper by A1,WAYBEL_0:25,YELLOW_9:47;
  hence thesis by PRE_TOPC:def 2;
end;
