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 Th34:
  seq is summable & (for n holds seq.n>0) implies Sum(seq)>0
proof
  assume that
A1: seq is summable and
A2: for n holds seq.n>0;
A3: Sum(seq)=(Partial_Sums(seq).0)+Sum(seq^\(0+1)) by A1,SERIES_1:15
    .= seq.0+Sum(seq^\1) by SERIES_1:def 1;
A4: now
    let n;
    (seq^\1).n = seq.(1+n) by NAT_1:def 3;
    hence (seq^\1).n>=0 by A2;
  end;
  seq^\1 is summable by A1,SERIES_1:12;
  then Sum(seq^\1)>=0 by A4,SERIES_1:18;
  then Sum(seq)>0+0 by A2,A3,XREAL_1:8;
  hence thesis;
end;
