
theorem Th2:
  for f being FinSequence of F_Complex, g being FinSequence of REAL
st len f = len g & for i being Element of NAT st i in dom f holds |. f/.i .| =
  g.i holds |. Product f .| = Product g
proof
  defpred P[Nat] means for f being FinSequence of F_Complex, g
  being FinSequence of REAL st len f = len g & (for i being Element of NAT st
  i in dom f holds |. f/.i .| = g.i) & $1 = len f holds
  |. Product f .| = Product g;
  set FC = F_Complex;
  set cFC = the carrier of FC;
A1: for i being Nat st P[i] holds P[i+1]
  proof
    let i being Nat such that
A2: P[i];
    let f be FinSequence of FC, g be FinSequence of REAL such that
A3: len f = len g and
A4: for i being Element of NAT st i in dom f holds |. f/.i .| = g.i and
A5: i+1 = len f;
    consider g1 being FinSequence of REAL, r being Element of REAL such that
A6: g = g1^<*r*> by A3,A5,FINSEQ_2:19;
A7: len g = len g1 + len <*r*> by A6,FINSEQ_1:22
      .= len g1 + 1 by FINSEQ_1:39;
    then
A8: g.len f = r by A3,A6,FINSEQ_1:42;
    consider f1 being FinSequence of FC, c being Element of cFC such that
A9: f = f1^<*c*> by A5,FINSEQ_2:19;
A10: len f = len f1 + len <*c*> by A9,FINSEQ_1:22
      .= len f1 + 1 by FINSEQ_1:39;
    then
A11: dom f1 = dom g1 by A3,A7,FINSEQ_3:29;
A12: now
      let i be Element of NAT;
A13:  dom f1 c= dom f by A9,FINSEQ_1:26;
      assume
A14:  i in dom f1;
      then f1/.i = f1.i by PARTFUN1:def 6
        .= f.i by A9,A14,FINSEQ_1:def 7
        .= f/.i by A14,A13,PARTFUN1:def 6;
      hence |. f1/.i .| = g.i by A4,A14,A13
        .= g1.i by A6,A11,A14,FINSEQ_1:def 7;
    end;
    reconsider Pf1 = Product f1 as Element of COMPLEX by COMPLFLD:def 1;
A15: Product g = Product g1 * r by A6,RVSUM_1:96;
    reconsider cc = c as Element of COMPLEX by COMPLFLD:def 1;
    f <> {} by A5;
    then
A16: len f in dom f by FINSEQ_5:6;
    then f/.len f = f.len f by PARTFUN1:def 6
      .= c by A9,A10,FINSEQ_1:42;
    then
A17: |.cc.| = r by A4,A16,A8;
    Product f = Product f1 * c by A9,GROUP_4:6;
    hence |. Product f .| = |. Pf1 .|*|.cc.| by COMPLEX1:65
      .= Product g by A2,A3,A5,A15,A10,A7,A12,A17;
  end;
A18: P[0]
  proof
    let f be FinSequence of F_Complex, g be FinSequence of REAL such that
A19: len f = len g and
    for i being Element of NAT st i in dom f holds |. f/.i .| = g.i and
A20: 0 = len f;
A21:  f = <*>(the carrier of F_Complex) by A20;
    then
A22:  g = <*>(the carrier of F_Complex) by A19;
    Product f = 1r by A21,COMPLFLD:8,GROUP_4:8;
    hence thesis by A22,COMPLEX1:48,RVSUM_1:94;
  end;
A23: for i being Nat holds P[i] from NAT_1:sch 2(A18,A1);
  let f be FinSequence of F_Complex, g be FinSequence of REAL;
  assume len f = len g & for i being Element of NAT st i in dom f holds |. f
  /.i .| = g.i;
  hence thesis by A23;
end;
