
theorem RSSPAC2:
  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 being 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 A0,TH11
      .= (v1 + w1) . n by VALUED_1:1;
    end;
    hence v + w = v1 + w1;
  end;
