
theorem Th4:
  for f being FinSequence of F_Complex st for i being Element of
  NAT st i in dom f holds f.i is integer holds Sum f is integer
proof
  set FC = F_Complex;
  let f be FinSequence of FC;
  assume
A1: for i being Element of NAT st i in dom f holds f.i is integer;
  defpred P[Nat] means for f being FinSequence of FC st len f = $1
& for i being Element of NAT st i in dom f holds f.i is integer holds Sum f is
  integer;
A2: for n being Nat st P[n] holds P[n+1]
  proof
    let n be Nat such that
A3: P[n];
    let f being FinSequence of FC;
    assume that
A4: len f = n+1 and
A5: for i being Element of NAT st i in dom f holds f.i is integer;
    consider g being FinSequence of FC, c being Element of FC such that
A6: f = g^<*c*> by A4,FINSEQ_2:19;
A7: now
      let i be Element of NAT;
A8:   dom g c= dom f by A6,FINSEQ_1:26;
      assume
A9:   i in dom g;
      then f.i = g.i by A6,FINSEQ_1:def 7;
      hence g.i is integer by A5,A9,A8;
    end;
    0+1 <= len f by A4,NAT_1:13;
    then len f in dom f by FINSEQ_3:25;
    then
A10: f.(len f) is integer by A5;
    reconsider Sgc = Sum g, cc = c as Element of COMPLEX by COMPLFLD:def 1;
    len f = len g + len <*c*> by A6,FINSEQ_1:22
      .= len g +1 by FINSEQ_1:40;
    then reconsider Sgi = Sgc, ci = cc as Integer by A3,A4,A6,A7,A10,
FINSEQ_1:42;
    Sum f = Sum g + Sum <*c*> by A6,RLVECT_1:41
      .= Sgi + ci by RLVECT_1:44;
    hence thesis;
  end;
A11: len f is Element of NAT;
A12: P[0]
  proof
    let f be FinSequence of FC;
    assume len f = 0;
    then f = <*>(the carrier of F_Complex);
    hence thesis by COMPLFLD:7,RLVECT_1:43;
  end;
  for n being Nat holds P[n] from NAT_1:sch 2(A12,A2);
  hence thesis by A1,A11;
end;
