reserve a,b,i,j,k,l,m,n for Nat;

theorem
  for a,b,c,d be Real, n be Nat holds
    (a*b,c*d) In_Power n =
      (Newton_Coeff n) (#) ((a,c) Subnomial n) (#) ((b,d) Subnomial n)
  proof
    let a,b,c,d be Real, n be Nat;
    (a*b,c*d) In_Power n = (Newton_Coeff n) (#) ((a*b,c*d) Subnomial n) by INS
    .= (Newton_Coeff n) (#) (((a,c) Subnomial n) (#) ((b,d) Subnomial n))
      by ST;
    hence thesis by RFUNCT_1:9;
  end;
