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 Th37:
  z <> 1r & (for n holds seq.(n+1) = z * seq.n) implies for n
  holds Partial_Sums(seq).n = seq.0 * ((1r - z |^ (n+1))/(1r-z))
proof
  assume that
A1: z <> 1r and
A2: for n holds seq.(n+1) = z * seq.n;
  defpred P[Nat] means seq.$1=seq.0 * (z GeoSeq).$1;
A3: now
    let n be Nat;
    assume P[n];
    then seq.(n+1)=z * (seq.0 * (z GeoSeq).n) by A2
      .=seq.0 * (z * (z GeoSeq).n)
      .=seq.0 * (z GeoSeq).(n+1) by Def1;
    hence P[n+1];
  end;
  let n;
  seq.0 = seq.0 * 1r by COMPLEX1:def 4
    .= seq.0 * (z GeoSeq).0 by Def1;
  then
A4: P[0];
  for n being Nat holds P[n] from NAT_1:sch 2(A4,A3);
  then for n being Element of NAT holds P[n];
  then Partial_Sums(seq)=Partial_Sums(seq.0 (#) (z GeoSeq)) by VALUED_1:7
    .=seq.0 (#) Partial_Sums(z GeoSeq) by Th29;
  hence Partial_Sums(seq).n=seq.0 * Partial_Sums(z GeoSeq).n by VALUED_1:6
    .=seq.0 * ( (1r - z |^ (n+1))/(1r -z) ) by A1,Th36;
end;
