
theorem ::::
  for F being FinSequence of REAL st (for k being Element of NAT
  st k in dom F holds 0 <= F.k) holds 0 <= Product F
  proof
    defpred P[Nat] means for F being FinSequence of REAL st (for k
    being Element of NAT st k in dom F holds 0 <= F.k)
    & len F = $1 holds 0 <= Product F;
    let F be FinSequence of REAL;
    A1: for n being Nat st P[n] holds P[n+1]
    proof
      let n be Nat;
      assume
      A2: P[n];
      for F being FinSequence of REAL st (for k being Element of NAT st k in
      dom F holds 0 <= F.k) & len F = n+1 holds 0 <= Product F
      proof
        let F be FinSequence of REAL;
        assume
        A3: for k being Element of NAT st k in dom F holds 0 <= F.k;
        assume
        A4: len F = n+1; then
        consider F1,F2 being FinSequence of REAL such that
        A5: len F1 = n and
        A6: len F2 = 1 and
        A7: F = F1^F2 by FINSEQ_2:23;
        1 in Seg 1; then
        1 in dom F2 by A6,FINSEQ_1:def 3; then
        A8: F.(n+1) = F2.1 by A5,A7,FINSEQ_1:def 7;
        for k being Element of NAT st k in dom F1 holds 0 <= F1.k
        proof
          let k be Element of NAT;
          assume
          A9: k in dom F1; then
          0 <= F.k by A3,A7,FINSEQ_2:15;
          hence thesis by A7,A9,FINSEQ_1:def 7;
        end; then
        A10: 0 <= Product F1 by A2,A5;
        set x = F2.1;
        Seg (n+1) = dom F by A4,FINSEQ_1:def 3; then
        A11: 0 <= x by A3,A8,FINSEQ_1:3;
        Product F = Product (F1^<*x*>) by A6,A7,FINSEQ_1:40
        .= Product F1 * x by RVSUM_1:96;
        hence thesis by A10,A11;
      end;
      hence thesis;
    end;
    A12: P[0]
    proof let F be FinSequence of REAL such that
      for k being Element of NAT st k in dom F holds 0 <= F.k;
      assume len F = 0; then
      F = <*>REAL;
      hence thesis by RVSUM_1:94;
    end;
    A13: for n being Nat holds P[n] from NAT_1:sch 2(A12,A1);
    A14: ex n being Element of NAT st n = len F;
    assume for k being Element of NAT st k in dom F holds 0 <= F.k;
    hence thesis by A13,A14;
  end;
