reserve T for non empty TopSpace,
  A, B for Subset of T,
  F, G for Subset-Family of T;

theorem
  A is closed iff Der A c= A
proof
  hereby
    assume A is closed;
    then
A1: A = Cl A by PRE_TOPC:22;
    Cl A = A \/ Der A by TOPGEN_1:29;
    hence Der A c= A by A1,XBOOLE_1:7;
  end;
  assume
A2: Der A c= A;
  Cl A = A \/ Der A by TOPGEN_1:29
    .= A by A2,XBOOLE_1:12;
  hence thesis;
end;
