
theorem Th41:
  for F being FinSequence of REAL st
  (for k being Element of NAT st k in dom F holds F.k > 0) holds Product F > 0
proof
  defpred P[Nat] means for F being FinSequence of REAL st (for k
being Element of NAT st k in dom F holds F.k > 0) & len F = $1 holds Product F
  > 0;
  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 F.k > 0) & len F = n+1 holds Product F > 0
    proof
      let F be FinSequence of REAL;
      assume
A3:   for k being Element of NAT st k in dom F holds F.k > 0;
      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 by FINSEQ_1:3;
      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 F1.k > 0
      proof
        let k be Element of NAT;
        assume
A9:     k in dom F1;
        then F.k > 0 by A3,A7,FINSEQ_2:15;
        hence thesis by A7,A9,FINSEQ_1:def 7;
      end;
      then
A10:  Product F1 > 0 by A2,A5;
      set x=F2.1;
      Seg (n+1) = dom F by A4,FINSEQ_1:def 3;
      then
A11:  x > 0 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 F.k > 0;
    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 F.k > 0;
  hence thesis by A13,A14;
end;
