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

theorem
  F = { A } implies Der F = { Der A }
proof
  assume
A1: F = { A };
  thus Der F c= { Der A }
  proof
    let x be object;
    assume
A2: x in Der F;
    then reconsider B = x as Subset of T;
    consider C being Subset of T such that
A3: B = Der C and
A4: C in F by A2,Def6;
    C = A by A1,A4,TARSKI:def 1;
    hence thesis by A3,TARSKI:def 1;
  end;
  let x be object;
  assume x in { Der A };
  then
A5: x = Der A by TARSKI:def 1;
  A in F by A1,TARSKI:def 1;
  hence thesis by A5,Def6;
end;
