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;

theorem Th3:
  (for n holds rseq.n = 0) implies rseq is absolutely_summable
proof
  assume
A1: for n holds rseq.n = 0;
  take 0;
  let p be Real such that
A2: 0<p;
  take 0;
  let m be Nat such that
  0<=m;
  |.(Partial_Sums abs(rseq)).m-0.| = |.0-0.| by A1,Th2
    .= 0 by ABSVALUE:def 1;
  hence |.(Partial_Sums abs(rseq)).m-0.|<p by A2;
end;
