reserve s for set,
  i,j for natural Number,
  k for Nat,
  x,x1,x2,x3 for Real,
  r,r1,r2,r3,r4 for Real,
  F,F1,F2,F3 for real-valued FinSequence,
  R,R1,R2 for Element of i-tuples_on REAL;

theorem
  for F being complex-valued FinSequence holds
  (ex k st k in dom F & F.k = 0) iff Product F = 0
proof
  defpred P[Nat] means for F being complex-valued FinSequence st len F = $1
  holds (ex k st k in Seg $1 & F.k = 0) iff Product F = 0;
  let F be complex-valued FinSequence;
A1: Seg len F = dom F by FINSEQ_1:def 3;
A2: for i be Nat st P[i] holds P[i+1]
  proof
    let i be Nat such that
A3: for F being complex-valued FinSequence st len F = i holds (ex k
    st k in Seg i & F.k = 0) iff Product F = 0;
    let F be complex-valued FinSequence;
    assume
A4: len F = i+1;
    then consider
    F9 being complex-valued FinSequence, x being Complex
    such that
A5: F = F9^<*x*> by Lm4;
A6: len F = len F9 + 1 by A5,FINSEQ_2:16;
A7: Product F = Product F9 * x by A5,Th96;
    thus (ex k st k in Seg (i+1) & F.k = 0) implies Product F = 0
    proof
      given k such that
A8:   k in Seg (i+1) and
A9:   F.k = 0;
      now
        per cases by A8,FINSEQ_2:7;
        suppose
A10:      k in Seg i;
          Seg len F9 = dom F9 by FINSEQ_1:def 3;
          then F9.k = F.k by A4,A5,A6,A10,FINSEQ_1:def 7;
          then Product F9 = 0 by A3,A4,A6,A9,A10;
          hence thesis by A7;
        end;
        suppose
          k = i+1;
          then x = 0 by A4,A5,A6,A9,FINSEQ_1:42;
          hence thesis by A7;
        end;
      end;
      hence thesis;
    end;
    assume
A11: Product F = 0;
    per cases by A7,A11;
    suppose
      Product F9 = 0;
      then consider k such that
A12:  k in Seg i and
A13:  F9.k = 0 by A3,A4,A6;
      Seg len F9 = dom F9 by FINSEQ_1:def 3;
      then F.k = 0 by A4,A5,A6,A12,A13,FINSEQ_1:def 7;
      hence thesis by A12,FINSEQ_2:8;
    end;
    suppose
      x = 0;
      then F.(i+1) = 0 by A4,A5,A6,FINSEQ_1:42;
      hence thesis by FINSEQ_1:4;
    end;
  end;
A14: P[0] proof let F be complex-valued FinSequence;
   assume len F = 0;
    then F = <*>COMPLEX;
   hence thesis by Th94;
  end;
  for i be Nat holds P[i] from NAT_1:sch 2(A14,A2);
  hence thesis by A1;
end;
