reserve X for non empty set,
  S for SigmaField of X,
  M for sigma_Measure of S,
  E for Element of S,
  F,G for Functional_Sequence of X,ExtREAL,
  I for ExtREAL_sequence,
  f,g for PartFunc of X,ExtREAL,
  seq, seq1, seq2 for ExtREAL_sequence,
  p for ExtReal,
  n,m for Nat,
  x for Element of X,
  z,D for set;

theorem
  (for n be Nat holds 0 < seq.n) implies for m be Nat holds 0 < (
  Partial_Sums seq).m
proof
  defpred P[Nat] means 0 < (Partial_Sums seq).$1;
  assume
A1: for n be Nat holds 0 < seq.n;
A2: for k be Nat st P[k] holds P[k+1]
  proof
    let k be Nat;
    assume
A3: P[k];
A4: (Partial_Sums seq).(k+1) = (Partial_Sums seq).k + seq.(k+1) by Def1;
    seq.(k+1) > 0 by A1;
    hence thesis by A3,A4;
  end;
  (Partial_Sums seq).0 = seq.0 by Def1;
  then
A5: P[ 0 ] by A1;
  thus for m be Nat holds P[m] from NAT_1:sch 2(A5,A2);
end;
