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 Th32:
  X common_on_dom H implies for x st x in X holds (r(#)H)#x = r*(H#x)
  proof
    assume
    A1: X common_on_dom H;

    let x;
    assume x in X;
    then {x} common_on_dom H by A1, Th25;
    hence thesis by Th29;
  end;
