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 Th16:
  seq is nonnegative implies Partial_Sums seq is nonnegative &
  Partial_Sums seq is non-decreasing
proof
  set PS = Partial_Sums seq;
  defpred P[Nat] means 0 <= PS.$1;
  assume
A1: seq is nonnegative;
A2: for k be Nat st P[k] holds P[k+1]
  proof
    let k be Nat;
    assume
A3: P[k];
A4: PS.(k+1) = PS.k + seq.(k+1) by Def1;
    seq.(k+1) >= 0 by A1,SUPINF_2:51;
    hence thesis by A3,A4;
  end;
  PS.0 = seq.0 by Def1;
  then
A5: P[ 0 ] by A1,SUPINF_2:51;
  for m be Nat holds P[m] from NAT_1:sch 2(A5,A2);
  then for k be object st k in dom PS holds 0 <= PS.k;
  hence PS is nonnegative by SUPINF_2:52;
  for n,m be Nat st m <= n holds (Partial_Sums seq).m <= (
  Partial_Sums seq).n
  proof
    let n,m be Nat;
    reconsider m1=m as Nat;
    defpred Q[Nat] means PS.m1 <= PS.$1;
A6: for k be Nat holds PS.k <= PS.(k+1)
    proof
      let k be Nat;
A7:   0 <= seq.(k+1) by A1,SUPINF_2:51;
      PS.(k+1) = PS.k + seq.(k+1) by Def1;
      hence thesis by A7,XXREAL_3:39;
    end;
A8: for k be Nat st k >= m1 & (for l be Nat st l >= m1 & l < k holds Q[l])
    holds Q[k]
    proof
      let k be Nat;
      assume that
A9:   k >= m1 and
A10:  for l be Nat st l >= m1 & l < k holds Q[l];
      now
        assume k > m1;
        then
A11:    k >= m1 + 1 by NAT_1:13;
        per cases by A11,XXREAL_0:1;
        suppose
          k = m1 + 1;
          hence thesis by A6;
        end;
        suppose
A12:      k > m1 + 1;
          then reconsider l = k-1 as Element of NAT by NAT_1:20;
          k < k+1 by NAT_1:13;
          then
A13:      k > l by XREAL_1:19;
          k = l+1;
          then
A14:      PS.l <= PS.k by A6;
          l >= m1 by A12,XREAL_1:19;
          then PS.m1 <= PS.l by A10,A13;
          hence thesis by A14,XXREAL_0:2;
        end;
      end;
      hence thesis by A9,XXREAL_0:1;
    end;
A15: for k be Nat st k >= m1 holds Q[k] from NAT_1:sch 9(A8);
    assume m <= n;
    hence thesis by A15;
  end;
  hence thesis by RINFSUP2:7;
end;
