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
  |.z.| < 1 & (for n holds seq.(n+1) = z * seq.n) implies seq is
  summable & Sum(seq) = seq.0/(1r-z)
proof
  assume that
A1: |.z.|< 1 and
A2: for n holds seq.(n+1) = z * seq.n;
  now
    let n be Element of NAT;
    thus Partial_Sums(seq).n = seq.0 * ((1r - z |^ (n+1))/(1r-z)) by A1,A2,Th37
,COMPLEX1:48
      .=seq.0 * Partial_Sums(z GeoSeq).n by A1,Th36,COMPLEX1:48
      .=(seq.0 (#) Partial_Sums(z GeoSeq)).n by VALUED_1:6;
  end;
  then
A3: Partial_Sums(seq)=(seq.0 (#) Partial_Sums(z GeoSeq));
A4: z GeoSeq is summable by A1,Th64;
  hence seq is summable by A3;
  Sum seq=seq.0 * Sum(z GeoSeq) by A3,A4,COMSEQ_2:18
    .=seq.0 * (1r/(1r-z)) by A1,Th64
    .=seq.0 * 1r/(1r-z) by XCMPLX_1:74
    .=seq.0/(1r-z) by COMPLEX1:def 4;
  hence thesis;
end;
