
theorem TH12:
  for f,h be VECTOR of R_Algebra_of_Big_Oh_poly
  for f9,h9 be Function of NAT,REAL
  st f9=f & h9=h
  for a be Real holds h = a*f iff
  for x be Nat holds h9.x = a*f9.x
  proof
    let f,h be VECTOR of R_Algebra_of_Big_Oh_poly;
    let f9,h9 be Function of NAT,REAL such that
    A1: f9=f & h9=h;
    let a be Real;
    A0:a is Element of REAL by XREAL_0:def 1;
    reconsider f1=f, h1=h as VECTOR of RAlgebra NAT by LM12;
    A3: now
    assume
    A4: h = a*f;
    let x be Nat;
    LXN:x in NAT by ORDINAL1:def 12;
    h1=a*f1 by A4,LM17,A0;
    hence h9.x=a*f9.x by A1,FUNCSDOM:4,LXN;
  end;
  now
    assume for x be Element of NAT holds h9.x=a*f9.x;
    then h1=a*f1 by A1,FUNCSDOM:4;
    hence h =a*f by A0,LM17;
  end;
  hence thesis by A3;
end;
