reserve D for non empty set,
  D1,D2,x,y for set,
  n,k for Nat,
  p,x1 ,r for Real,
  f for Function;
reserve F for Functional_Sequence of D1,D2;
reserve G,H,H1,H2,J for Functional_Sequence of D,REAL;

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 RFUNCT_1:17
      .=r(#)(p(#)H).n by Def1
      .=(r(#)(p(#)H)).n by Def1;
  end;
  hence thesis by FUNCT_2:63;
end;
