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

theorem Th41:
  for T being TopSpace, A being Subset of T holds A is perfect iff Der A = A
proof
  let T be TopSpace, A be Subset of T;
  thus A is perfect implies Der A = A by Lm1;
  assume
A1: Der A = A;
A2: Cl A = Der A \/ A by Th29
    .= A by A1;
  A is dense-in-itself by A1;
  hence thesis by A2;
end;
