
theorem Th14:
  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
  defpred P[Nat] means for F being FinSequence of F_Complex 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;
  let G be FinSequence of F_Complex such that
A1: for i being Element of NAT st i in dom G holds G.i is Integer;
A2: for k being Nat st P[k] holds P[k+1]
  proof
    let k be Nat such that
A3: P[k];
    let F be FinSequence of F_Complex such that
A4: len F = k+1 and
A5: for i being Element of NAT st i in dom F holds F.i is Integer;
    F <> {} by A4;
    then consider G being FinSequence, x being object such that
A6: F = G^<*x*> by FINSEQ_1:46;
    len F in Seg len F by A4,FINSEQ_1:3;
    then
A7: len F in dom F by FINSEQ_1:def 3;
    reconsider f2=<*x*> as FinSequence of F_Complex by A6,FINSEQ_1:36;
    reconsider f1=G as FinSequence of F_Complex by A6,FINSEQ_1:36;
    rng f2 c= the carrier of F_Complex by FINSEQ_1:def 4;
    then {x} c= the carrier of F_Complex by FINSEQ_1:38;
    then reconsider m=x as Element of F_Complex by ZFMISC_1:31;
    k + 1 = len f1 + len f2 by A4,A6,FINSEQ_1:22;
    then
A8: k + 1 = len f1 + 1 by FINSEQ_1:39;
    then F.(len F) = m by A4,A6,FINSEQ_1:42;
    then reconsider i2 = m as Integer by A5,A7;
    for j being Element of NAT st j in dom f1 holds f1.j is Integer
    proof
A9:   dom f1 c= dom F by A6,FINSEQ_1:26;
      let j be Element of NAT such that
A10:  j in dom f1;
      F.j = f1.j by A6,A10,FINSEQ_1:def 7;
      hence thesis by A5,A10,A9;
    end;
    then reconsider i1 = Sum f1 as Integer by A3,A8;
    Sum F = Sum f1 + m by A6,FVSUM_1:71;
    then Sum F = i1 + i2;
    hence thesis;
  end;
A11: P[0]
  proof
    let F be FinSequence of F_Complex such that
A12: len F = 0 and
    for i being Element of NAT st i in dom F holds F.i is Integer;
    0-tuples_on the carrier of F_Complex = {{}} & F = {} by A12,COMPUT_1:5;
    then F is Element of 0-tuples_on the carrier of F_Complex by TARSKI:def 1;
    hence thesis by COMPLFLD:7,FVSUM_1:74;
  end;
A13: for k being Nat holds P[k] from NAT_1:sch 2(A11,A2);
  len G = len G;
  hence thesis by A1,A13;
end;
