reserve rseq, rseq1, rseq2 for Real_Sequence;
reserve seq, seq1, seq2 for Complex_Sequence;
reserve k, n, n1, n2, m for Nat;
reserve p, r for Real;
reserve z for Complex;
reserve Nseq,Nseq1 for increasing sequence of NAT;

theorem Th43:
  (0 < |.z.| & |.z.| < 1 & seq.0 = z & for n holds seq.(n+1) = seq
  .n * z) implies seq is convergent & lim(seq)=0c
proof
  assume that
A1: 0 < |.z.| and
A2: |.z.| < 1;
  deffunc g(Nat) = |.z.| to_power ($1+1);
  consider rseq1 such that
A3: for n holds rseq1.n=g(n) from SEQ_1:sch 1;
  assume that
A4: seq.0 = z and
A5: for n holds seq.(n+1) = seq.n * z;
A6: for n holds |.seq.n.| = |.z.| to_power (n+1)
  proof
    defpred P[Nat] means |. seq.$1 .|= |.z.| to_power ($1+1);
A7: for k being Nat st P[k] holds P[k+1]
    proof
      let k be Nat;
      assume
A8:   P[k];
      |.seq.(k+1).| = |.seq.k * z.| by A5
        .= |.z.| to_power (k+1) * |.z.| by A8,COMPLEX1:65
        .= |.z.| to_power (k+1) * |.z.| to_power 1 by POWER:25
        .= |.z.| to_power ((k+1)+1) by A1,POWER:27;
      hence thesis;
    end;
A9: P[0] by A4,POWER:25;
    for n holds P[n] from NAT_1:sch 2(A9,A7);
    hence thesis;
  end;
A10: now
    let n;
    abs(Re seq).n = |.Re seq.n.| by SEQ_1:12;
    then
A11: abs(Re seq).n = |.Re(seq.n).| by Def5;
    |.Re(seq.n).| <= |.seq.n.| & |.seq.n.| = |.z.| to_power (n+1) by A6,Th13;
    hence abs(Re seq).n <= rseq1.n by A3,A11;
  end;
A12: now
    let n;
A13: |.seq.n.| = |.z.| to_power (n+1) by A6;
    abs(Im seq).n = |.Im seq.n.| & |.Im(seq.n).| <= |.seq.n.| by Th13,
SEQ_1:12;
    then abs(Im seq).n <= |.z.| to_power (n+1) by A13,Def6;
    hence abs(Im seq).n <= rseq1.n by A3;
  end;
  C.0=0;
  then
A14: lim(C)=0 by SEQ_4:25;
A15: rseq1 is convergent & lim(rseq1)=0 by A1,A2,A3,SERIES_1:1;
  now
    let n;
    abs(Re seq).n=|.Re seq.n.| by SEQ_1:12;
    then 0 <= abs(Re seq).n by COMPLEX1:46;
    hence C.n <= abs(Re seq).n;
  end;
  then
A16: abs(Re seq) is convergent & lim(abs(Re seq))=0 by A14,A15,A10,SEQ_2:19,20;
  then
A17: Re seq is convergent by SEQ_4:15;
  now
    let n;
    abs(Im seq).n=|.Im seq.n.| by SEQ_1:12;
    then 0 <= abs(Im seq).n by COMPLEX1:46;
    hence C.n <= abs(Im seq).n;
  end;
  then
A18: abs(Im seq) is convergent & lim(abs(Im seq))=0 by A14,A15,A12,SEQ_2:19,20;
  then
A19: Im seq is convergent by SEQ_4:15;
  lim(Im seq)=0 by A18,SEQ_4:15;
  then
A20: Im lim(seq)=0 by A17,A19,Th42;
  lim(Re seq)=0 by A16,SEQ_4:15;
  then Re lim(seq)=0 by A17,A19,Th42;
  hence thesis by A17,A19,A20,Lm1,Th42,COMPLEX1:13;
end;
