
theorem ::NAT454:
  for f1,f2 being FinSequence of REAL st len f1 = len f2 &
  (for k be Element of NAT
  st k in dom f2 holds f1.k>=f2.k & f2.k>=0) holds Product f1 >= Product f2
  proof
    let f1,f2 be 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 & f2.k >= 0);
    for r be Real st r in rng f2 holds r >= 0
    proof
      let r be Real such that
      B1: r in rng f2;
      consider k be object such that
      B2: k in dom f2 & f2.k = r by B1,FUNCT_1:def 3;
      reconsider k as Element of NAT by B2;
      thus thesis by A1,B2;
    end; then
    reconsider f2 as nonnegative-yielding FinSequence of REAL
      by PARTFUN3:def 4;
    for r be Real st r in rng f1 holds r >= 0
    proof
      let r be Real such that
      B1: r in rng f1;
      consider k be object such that
      B2: k in dom f1 & f1.k = r by B1,FUNCT_1:def 3;
      reconsider k as Element of NAT by B2;
      dom f1 = dom f2 by A1,FINSEQ_3:29; then
      f1.k >= f2.k & f2.k >= 0 by A1,B2;
      hence thesis by B2;
    end; then
    reconsider f1 as nonnegative-yielding FinSequence of REAL
      by PARTFUN3:def 4;
    len f1 = len f2 &
      (for k be Element of NAT st k in dom f2 holds f1.k>=f2.k) by A1;
    hence thesis by N454;
  end;
