
theorem Th51:
  for S being complete LATTICE, T being Scott TopAugmentation of S
  holds the topology of T = sigma S
proof
  let S be complete LATTICE;
  let T be Scott TopAugmentation of S;
  set R = TopRelStr(#the carrier of S, the InternalRel of S, sigma S#);
  the RelStr of R = the RelStr of S;
  then reconsider R as TopAugmentation of S by Def4;
  reconsider R as correct Scott TopAugmentation of S by Th48,Th49;
A1: the RelStr of T = the RelStr of R by Def4;
  thus the topology of T c= sigma S
  proof
    let x be object;
    assume
A2: x in the topology of T;
    then reconsider A = x as Subset of T;
    reconsider C = A as Subset of R by A1;
    A is open by A2;
    then C is open by A1,Th50;
    hence thesis;
  end;
  let x be object;
  assume
A3: x in sigma S;
  then reconsider A = x as Subset of R;
  reconsider C = A as Subset of T by A1;
  A is open by A3;
  then C is open by A1,Th50;
  hence thesis;
end;
