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 Th37:
  F is with_the_same_dom & x in dom(F.0) & (for k be Nat holds F.k
is nonnegative) & n <= m implies ((Partial_Sums F).n).x <= ((Partial_Sums F).m)
  .x
proof
  assume
A1: F is with_the_same_dom;
  set PF = Partial_Sums F;
  assume
A2: x in dom(F.0);
  defpred P[Nat] means (PF.n).x <= (PF.$1).x;
  assume
A3: for m be Nat holds F.m is nonnegative;
A4: for k be Nat holds (PF.k).x <= (PF.(k+1)).x
  proof
    let k be Nat;
A5: PF.(k+1) = PF.k + F.(k+1) by Def4;
    F.(k+1) is nonnegative by A3;
    then
A6: 0. <= (F.(k+1)).x by SUPINF_2:39;
    dom(PF.(k+1)) = dom(F.0) by A1,A3,Th29,Th30;
    then (PF.(k+1)).x = (PF.k).x + (F.(k+1)).x by A2,A5,MESFUNC1:def 3;
    hence thesis by A6,XXREAL_3:39;
  end;
A7: for k be Nat st k >= n & (for l be Nat st l >= n & l < k holds P[l])
  holds P[k]
  proof
    let k be Nat;
    assume that
A8: k >= n and
A9: for l be Nat st l >= n & l < k holds P[l];
    now
A10:  now
        assume
A11:    k > n+1;
        then reconsider l = k-1 as Element of NAT by NAT_1:20;
        k < k+1 by NAT_1:13;
        then
A12:    k > l by XREAL_1:19;
        k = l+1;
        then
A13:    (PF.l).x <= (PF.k).x by A4;
        l >= n by A11,XREAL_1:19;
        then (PF.n).x <= (PF.l).x by A9,A12;
        hence thesis by A13,XXREAL_0:2;
      end;
      assume k > n;
      then k >= n + 1 by NAT_1:13;
      then k = n+1 or k > n+1 by XXREAL_0:1;
      hence thesis by A4,A10;
    end;
    hence thesis by A8,XXREAL_0:1;
  end;
A14: for k being Nat st k >= n holds P[k] from NAT_1:sch 9(A7);
  assume n <= m;
  hence thesis by A14;
end;
