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

theorem
  for a,b be Real, n be Nat holds
    (a,b) In_Power n = ((a,1) In_Power n) (#) ((1,b) Subnomial n) &
    (a,b) In_Power n = ((a,1) Subnomial n) (#) ((1,b) In_Power n)
  proof
    let a,b be Real, n be Nat;
    A1: (a,b) In_Power n = (Newton_Coeff n) (#) ((a,b) Subnomial n) by INS
    .= (Newton_Coeff n) (#) (((a,1) Subnomial n) (#) ((1,b) Subnomial n))
      by DAB;
    (Newton_Coeff n) (#) ((a,1) Subnomial n) = ((a,1) In_Power n) &
    (Newton_Coeff n) (#) ((1,b) Subnomial n) = ((1,b) In_Power n) by INS;
    hence thesis by A1,F123;
  end;
