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
  rseq1 is summable & (ex m st for n st m<=n holds |.seq2.n.| <= rseq1.n
  ) implies seq2 is absolutely_summable
proof
  assume that
A1: rseq1 is summable and
A2: ex m st for n st m<=n holds |.seq2.n.| <= rseq1.n;
  consider m such that
A3: for n st m<=n holds |.seq2.n.| <= rseq1.n by A2;
A4: now
    let n;
    |.seq2.n.|=|.seq2.|.n by VALUED_1:18;
    hence 0 <= |.seq2.|.n by COMPLEX1:46;
  end;
  now
    let n;
    assume m <= n;
    then |.seq2.n.| <= rseq1.n by A3;
    hence |.seq2.|.n <= rseq1.n by VALUED_1:18;
  end;
  hence |.seq2.| is summable by A1,A4,SERIES_1:19;
end;
