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 Th63:
  seq is absolutely_summable implies seq is summable
proof
  assume
A1: seq is absolutely_summable;
  now
    let p;
    assume 0<p;
    then consider n such that
A2: for m st n <= m holds |.Partial_Sums(|.seq.|).m-Partial_Sums(|.
    seq.|).n.|< p by A1,SEQ_4:41;
    take n;
    let m;
    assume n <= m;
    then
A3: |.Partial_Sums(|.seq.|).m-Partial_Sums(|.seq.|).n.|< p by A2;
    |.Partial_Sums(seq).m-Partial_Sums(seq).n.| <= |.Partial_Sums(|.seq
    .|).m-Partial_Sums(|.seq.|).n.| by Th31;
    hence |.Partial_Sums(seq).m-Partial_Sums(seq).n.| <p by A3,XXREAL_0:2;
  end;
  hence thesis by Th62;
end;
