
theorem LM17:
  for f be Point of R_Algebra_of_Big_Oh_poly,
  f1 be Point of RAlgebra NAT,
  a be Element of REAL st f=f1 holds
  a*f = a*f1
  proof
    let f be Point of R_Algebra_of_Big_Oh_poly,
    f1 be Point of RAlgebra NAT,
    a be Element of REAL;
    assume A1: f=f1;
    set X = Big_Oh_poly;
    set S = R_Algebra_of_Big_Oh_poly;
    A3: the carrier of R_Algebra_of_Big_Oh_poly
    = X by defAlgebra;
    thus a*f = ((RealFuncExtMult(NAT)) | [:REAL,X:]). ( [a,f] ) by defAlgebra
    .=a*f1 by A1,A3,
    FUNCT_1:49;
  end;
