
theorem
  for a be positive Real, f be nonnegative-yielding FinSequence of REAL st
    (for k be Element of NAT st k in dom f holds f.k <= a)
  holds Product (f) <= a|^(len f)
  proof
    let a be positive Real, f be nonnegative-yielding FinSequence of REAL
      such that
    A1: for k be Element of NAT st k in dom f holds f.k <= a;
    a in REAL by XREAL_0:def 1; then
    reconsider g = ((len f)|-> a) as FinSequence of REAL by FINSEQ_2:63;
    reconsider g as positive-yielding FinSequence of REAL;
    A2: (len f = len g);
    for k be Element of NAT st k in dom f holds g.k >= f.k
    proof
      let k be Element of NAT such that
      B1: k in dom f;
      g.k = a by B1,Lmkdf;
      hence thesis by A1,B1;
    end;  then
    Product (f) <= Product (g) by A2,N454;
    hence thesis by NEWTON:def 1;
  end;
