
theorem LM15:
  for f,g be Point of R_Algebra_of_Big_Oh_poly,
  f1,g1 be Point of RAlgebra NAT st f=f1 & g=g1 holds
  f*g = f1*g1
  proof
    let f,g be Point of R_Algebra_of_Big_Oh_poly,
    f1,g1 be Point of RAlgebra NAT;
    assume A1: f=f1 & g=g1;
    set X = Big_Oh_poly;
    set S = R_Algebra_of_Big_Oh_poly;
    A3a: the carrier of R_Algebra_of_Big_Oh_poly = X by defAlgebra;
    thus f*g = ((RealFuncMult(NAT)) || X). ( [f,g] ) by defAlgebra
    .=f1*g1 by A1,A3a,FUNCT_1:49;
  end;
