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
  (for n st n <= m holds seq1.n = seq2.n) implies Partial_Sums(seq1).m =
  Partial_Sums(seq2).m
proof
  defpred P[Nat] means $1 <= m implies Partial_Sums(seq1).$1=
  Partial_Sums(seq2).$1;
  assume
A1: for n st n <= m holds seq1.n = seq2.n;
A2: for k be Nat st P[k] holds P[k+1]
  proof
    let k be Nat such that
A3: P[k];
    assume
A4: k+1 <= m;
    k < k+1 by XREAL_1:29;
    hence Partial_Sums(seq1).(k+1) =Partial_Sums(seq2).k+seq1.(k+1) by A3,A4,
SERIES_1:def 1,XXREAL_0:2
      .=Partial_Sums(seq2).k+seq2.(k+1) by A1,A4
      .=Partial_Sums(seq2).(k+1) by SERIES_1:def 1;
  end;
A5: P[0]
  proof
    assume 0 <= m;
    thus Partial_Sums(seq1).0=seq1.0 by SERIES_1:def 1
      .=seq2.0 by A1
      .=Partial_Sums(seq2).0 by SERIES_1:def 1;
  end;
  for k holds P[k] from NAT_1:sch 2(A5,A2);
  hence thesis;
end;
