reserve x, x1, x2, y, X, D for set,
  i, j, k, l, m, n, N for Nat,
  p, q for XFinSequence of NAT,
  q9 for XFinSequence,
  pd, qd for XFinSequence of D;
reserve pN, qN for Element of NAT^omega;
reserve seq1,seq2,seq3,seq4 for Real_Sequence,
  r,s,e for Real,
  Fr,Fr1, Fr2 for XFinSequence of REAL;

theorem Th52:
  for seq1, n, m st n <= m & for i holds seq1.i>=0 holds (
  Partial_Sums seq1).n <= (Partial_Sums seq1).m
proof
  let seq1, n, m such that
A1: n <= m and
A2: for i holds seq1.i>=0;
  set S=Partial_Sums seq1;
  defpred P[Nat] means S.n <= S.(n+$1);
A3: for i st P[i] holds P[i+1]
  proof
    let i such that
A4: P[i];
    set ni=n+i;
    S.(ni+1)=S.ni+seq1.(ni+1) & seq1.(ni+1)>=0 by A2,SERIES_1:def 1;
    then S.(ni+1) >= S.ni + (0 qua Nat) by XREAL_1:6;
    hence thesis by A4,XXREAL_0:2;
  end;
A5: P[0];
A6: for i holds P[i] from NAT_1:sch 2(A5,A3);
  reconsider m9=m, n9=n as Nat;
A7: n9+(m9-n9)=m9;
  m9-n9 is Nat by A1,NAT_1:21;
  hence thesis by A6,A7;
end;
