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 Th29:
  seq is summable & (for k holds seq.k>=0) implies Sum(seq)>=
  Partial_Sums(seq).K
proof
  assume that
A1: seq is summable and
A2: for k holds seq.k>=0;
A3: now
    let k;
    (seq^\(K+1)).k = seq.(K+1+k) by NAT_1:def 3;
    hence (seq^\(K+1)).k>=0 by A2;
  end;
  seq^\(K+1) is summable by A1,SERIES_1:12;
  then Sum(seq^\(K+1))>=0 by A3,SERIES_1:18;
  then Partial_Sums(seq).K+Sum(seq^\(K+1))>=Partial_Sums(seq).K+0 by XREAL_1:6;
  hence thesis by A1,SERIES_1:15;
end;
