reserve D for non empty set,
  D1, D2, x, y, Z for set,
  n, k for Nat,
  p, x1, r for Real,
  f for Function,
  Y for RealNormSpace,
  G, H, H1, H2, J for Functional_Sequence of D,the carrier of Y;
reserve
  x for Element of D,
  X for set,
  S1, S2 for sequence of Y,
  f for PartFunc of D,the carrier of Y;

theorem Th22:
  Z c= X & Z<>{} & X common_on_dom H implies Z common_on_dom H
  proof
    assume that
    A1: Z c= X and
    A2: Z<>{} and
    A3: X common_on_dom H;
    now
      let n;
      X c= dom (H.n) by A3;
      hence Z c= dom (H.n) by A1;
    end;
    hence thesis by A2;
  end;
