
theorem
for a be Real for f be FinSequence of REAL st
(for k be Element of NAT st k in dom f holds 0 < f.k <= a)
holds Product f <= Product ((len f) |-> a)
proof
  let a be Real;
  let f be FinSequence of REAL such that
  A0: for k be Element of NAT st k in dom f holds 0 < 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;
  A1: len f = len g;
  for k be Element of NAT st k in dom f holds  f.k <= g.k & f.k > 0
  proof
    let k be Element of NAT such that
    B1: k in dom f;
    g.k = a by B1,Lmkdf;
    hence thesis by A0,B1;
  end;
  hence thesis by A1,NAT_4:54;
end;
