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

theorem SSI:
  for a,b be positive Real, n be non zero Nat holds
    Sum ((a,b) Subnomial (n+1)) < Sum ((a,b) In_Power (n+1))
  proof
    let a,b be positive Real, n be non zero Nat;
    A1: for i be Nat holds
      ((a,b) Subnomial (n+1)).i <= ((a,b) In_Power (n+1)).i by ILS;
    reconsider h = ((a,b)Subnomial (n+1))
      as nonnegative-yielding FinSequence of REAL;
    reconsider g = ((a,b)In_Power (n+1))
      as nonnegative-yielding FinSequence of REAL;
    1+0 < 1+n by XREAL_1:6; then
    1*((a,b) Subnomial (n+1)).(n+1) <
      (Newton_Coeff (n+1)).(n+1)*((a,b) Subnomial (n+1)).(n+1)
        by XREAL_1:68; then
    ((a,b) Subnomial (n+1)).(n+1) <
      ((Newton_Coeff (n+1))(#)((a,b) Subnomial (n+1))).(n+1)
        by VALUED_1:5; then
    ((a,b) Subnomial (n+1)).(n+1) < ((a,b) In_Power (n+1)).(n+1) by INS;
    hence thesis by A1,NYS1;
  end;
