
theorem RSSPAC3:
  for a be Real,v being VECTOR of R_Algebra_of_Big_Oh_poly,
  v1 be Function of NAT,REAL
  st v=v1
  holds a*v= a (#) v1
  proof
    let a be Real;
    let v be VECTOR of R_Algebra_of_Big_Oh_poly,
    v1 be Function of NAT,REAL;
    assume A0: v=v1;
    a*v in the carrier of R_Algebra_of_Big_Oh_poly;
    then
    a*v in Big_Oh_poly by defAlgebra;then
    reconsider h=a*v as Function of NAT,REAL by DefX1;
    h = a (#) v1
    proof
      let n be Element of NAT;
      thus h. n = a* (v1 . n) by TH12,A0
      .= (a (#) v1) . n by VALUED_1:6;
    end;
    hence a*v = a (#) v1;
  end;
