reserve k, m, n, p, K, N for Nat;
reserve i for Integer;
reserve x, y, eps for Real;
reserve seq, seq1, seq2 for Real_Sequence;
reserve sq for FinSequence of REAL;

theorem Th15:
  for seq st seq^\1 is summable holds seq is summable & Sum(seq)=(
  seq.0)+Sum(seq^\1)
proof
  let seq;
  assume
A1: seq^\1 is summable;
  hence seq is summable by SERIES_1:13;
  thus Sum(seq) = Partial_Sums(seq).0+Sum(seq^\(1+0)) by A1,SERIES_1:13,15
    .= (seq.0)+Sum(seq^\1) by SERIES_1:def 1;
end;
