
theorem RSSPAC2A:
  for v, w being VECTOR of R_Algebra_of_Big_Oh_poly,
  v1,w1 being Function of NAT,REAL
  st v=v1 & w1=w
  holds v * w = v1 (#) w1
  proof
    let v, w be VECTOR of R_Algebra_of_Big_Oh_poly,
    v1,w1 be Function of NAT,REAL;
    assume A0: v=v1 & w1=w;
    v*w in the carrier of R_Algebra_of_Big_Oh_poly;
    then
    v*w in Big_Oh_poly by defAlgebra;then
    reconsider h=v*w as Function of NAT,REAL by DefX1;
    h = v1 (#) w1
    proof
      let n be Element of NAT;
      thus h. n = (v1 . n) * (w1 . n) by TH11A,A0
      .= (v1 (#) w1) . n by VALUED_1:5;
    end;
    hence v * w = v1 (#) w1;
  end;
