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

theorem Th33:
  for T being TopSpace, A being Subset of T st T is T_1 holds Cl Der A = Der A
proof
  let T be TopSpace, A be Subset of T;
  assume
A1: T is T_1;
  per cases;
  suppose
A2: T is non empty;
    Cl Der A = Der A \/ Der Der A by Th29;
    then
A3: Cl Der A c= Der A \/ Der A by A1,A2,Th32,XBOOLE_1:9;
    Der A c= Cl Der A by PRE_TOPC:18;
    hence thesis by A3;
  end;
  suppose
A4: T is empty;
    then Cl Der A = {};
    hence thesis by A4;
  end;
end;
