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 Th36:
  1r <> z implies for n holds Partial_Sums(z GeoSeq).n = (1r - z
  |^ (n+1))/(1r-z)
proof
  now
    let z;
    defpred P[Nat] means
Partial_Sums(z GeoSeq).$1 = (1r - z |^ ($1+1))/(1r-z);
    assume 1r <> z;
    then
A1: 1r-z <>0c;
A2: for n st P[n] holds P[n+1]
    proof
      let n;
      assume P[n];
      hence
      Partial_Sums(z GeoSeq).(n+1) = (1r - z |^ (n+1))/(1r-z)+ z |^ (n+1)
      * 1r by COMPLEX1:def 4,SERIES_1:def 1
        .= (1r - z |^ (n+1))/(1r-z)+ z |^ (n+1) * ((1r-z)/(1r-z)) by A1,
COMPLEX1:def 4,XCMPLX_1:60
        .= (1r - z |^ (n+1))/(1r-z)+ (z |^ (n+1) * (1r-z))/(1r-z) by
XCMPLX_1:74
        .= (1r - z |^ (n+1) + (z |^ (n+1) -z |^(n+1) * z ))/(1r-z) by
COMPLEX1:def 4,XCMPLX_1:62
        .= (1r - z |^(n+1) * z )/(1r-z)
        .= (1r - z |^ ((n+1) +1) )/(1r-z) by NEWTON:6;
    end;
    Partial_Sums(z GeoSeq).0 =(z GeoSeq).0 by SERIES_1:def 1
      .=1r by Def1
      .=(1r -1 * z )/(1r-z) by A1,COMPLEX1:def 4,XCMPLX_1:60
      .=(1r-z |^ (0+1) )/(1r-z);
    then
A3: P[0];
    thus for n holds P[n] from NAT_1:sch 2(A3,A2);
  end;
  hence thesis;
end;
