
theorem N454:
  for f1,f2 be nonnegative-yielding FinSequence of REAL st len f1 = len f2
  & (for k be Element of NAT st k in dom f2 holds f1.k>=f2.k) holds
  Product f1 >= Product f2
  proof
    let f1,f2 be nonnegative-yielding FinSequence of REAL such that
    A1: len f1 = len f2 & (for k be Element of NAT st k in dom f2 holds
    f1.k >= f2.k);
    per cases;
    suppose ex l be Element of NAT st l in dom f2 & f2.l = 0;
      hence thesis by RVSUM_1:103;
    end;
    suppose
      for l be Element of NAT st l in dom f2 holds f2.l <> 0; then
      for k be Element of NAT st k in dom f2 holds f1.k >= f2.k & f2.k > 0
        by A1,XXREAL_0:1;
      hence thesis by A1,NAT_4:54;
    end;
  end;
