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;
reserve x for Element of D;

theorem :: Th33:
  H1 is_point_conv_on X & H2 is_point_conv_on X
  implies for x st x in X holds H1#x + H2#x = (H1+H2)#x
  & H1#x - H2#x = (H1-H2)#x by Th30;
