
theorem Th21:
  for T being non empty TopSpace st T is Frechet holds for A,B
  being Subset of T holds Cl_Seq({}T) = {} & A c= Cl_Seq(A) & Cl_Seq(A \/ B) =
  Cl_Seq(A) \/ Cl_Seq(B) & Cl_Seq(Cl_Seq(A)) = Cl_Seq(A)
proof
  let T be non empty TopSpace;
  assume
A1: T is Frechet;
  let A,B be Subset of T;
  thus Cl_Seq({}T) = {} & A c= Cl_Seq(A) & Cl_Seq(A \/ B)=Cl_Seq(A) \/ Cl_Seq(
  B) by Th17,Th18,Th19;
  thus Cl_Seq(Cl_Seq(A)) = Cl_Seq(Cl(A)) by A1,Th20
    .= Cl(Cl(A)) by A1,Th20
    .= Cl_Seq(A) by A1,Th20;
end;
