reserve X for non empty set,
        x for Element of X,
        S for SigmaField of X,
        M for sigma_Measure of S,
        f,g,f1,g1 for PartFunc of X,REAL,
        l,m,n,n1,n2 for Nat,
        a,b,c for Real;
reserve k for positive Real;
reserve v,u for VECTOR of RLSp_LpFunct(M,k);
reserve v,u for VECTOR of RLSp_AlmostZeroLpFunct(M,k);
reserve x for Point of Pre-Lp-Space(M,k);
reserve x,y for Point of Lp-Space(M,k);

theorem Th68:
for F be Functional_Sequence of X,REAL, x be Element of X, n,m be Nat st
 F is with_the_same_dom & x in dom(F.0) &
(for k be Nat holds F.k is nonnegative) & n <= m holds
   ((Partial_Sums F).n).x <= ((Partial_Sums F).m).x
proof
   let F be Functional_Sequence of X,REAL, x be Element of X, n,m be Nat;
   assume A1: F is with_the_same_dom;
   assume A2: x in dom(F.0);
   assume A3: for m be Nat holds F.m is nonnegative;
   assume A4: n <= m;
   set PF = Partial_Sums F;
   defpred P[Nat] means (PF.n).x <= (PF.$1).x;
A5:for k be Nat holds (PF.k).x <= (PF.(k+1)).x
   proof
    let k be Nat;
A6:PF.(k+1) = PF.k + F.(k+1) by MESFUN9C:def 2;
A7:dom(PF.(k+1)) = dom(F.0) by A1,MESFUN9C:11;
    F.(k+1) is nonnegative & PF.k is nonnegative by A3,Th67; then
    0 <= (F.(k+1)).x & 0 <= (PF.k).x by MESFUNC6:51; then
    (PF.k).x + 0 <= (PF.k).x + (F.(k+1)).x by XREAL_1:7;
    hence thesis by A7,A2,A6,VALUED_1:def 1;
   end;
A8: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
A9: k >= n & for l be Nat st l >= n & l < k holds P[l];
    now assume k > n; then
     k >= n + 1 by NAT_1:13; then
A10: k = n+1 or k > n+1 by XXREAL_0:1;
     now assume A11: k > n+1; then
      reconsider l = k-1 as Nat by NAT_1:20;
      k < k+1 by NAT_1:13; then
      k > l & l >= n by A11,XREAL_1:19; then
A12:  (PF.n).x <= (PF.l).x by A9;
      k = l+1; then
      (PF.l).x <= (PF.k).x by A5;
      hence thesis by A12,XXREAL_0:2;
     end;
     hence thesis by A10,A5;
    end;
    hence thesis by A9,XXREAL_0:1;
   end;
   for k being Nat st k >= n holds P[k] from NAT_1:sch 9(A8);
   hence thesis by A4;
end;
