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

theorem NS:
  for a,b be Real, n be Nat holds
  (((a,b) In_Power n).1 = ((a,b) Subnomial n).1) &
    (((a,b) In_Power n).(n+1) = ((a,b) Subnomial n).(n+1))
  proof
    let a,b be Real, n be Nat;
    A1: ((a,b) Subnomial n).1 = ((a,b) In_Power n).1/(Newton_Coeff n).1
      by VALUED_1:17;
    ((a,b) Subnomial n).(n+1) = ((a,b) In_Power n).(n+1)/(Newton_Coeff n).(n+1)
      by VALUED_1:17;
    hence thesis by A1;
  end;
