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;

theorem
  (r*p)(#)H = r(#)(p(#)H)
  proof
    now
      let n be Element of NAT;
      thus ((r*p)(#)H).n=(r*p)(#)H.n by Def1
      .=r(#)(p(#)H.n) by VFUNCT_1:14
      .=r(#)(p(#)H).n by Def1
      .=(r(#)(p(#)H)).n by Def1;
    end;
    hence thesis by FUNCT_2:63;
  end;
