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 Th66:
for F be Functional_Sequence of X,REAL st
 (for m be Nat holds F.m in Lp_Functions(M,k))
 holds
  for m be Nat holds (Partial_Sums F).m in Lp_Functions(M,k)
proof
   let F be Functional_Sequence of X,REAL;
   assume A1: for m be Nat holds F.m in Lp_Functions(M,k);
   defpred P[Nat] means (Partial_Sums F).$1 in Lp_Functions(M,k);
   (Partial_Sums F).0 = F.0 by MESFUN9C:def 2; then
A2:P[ 0] by A1;
A3:now let j be Nat;
    assume P[j]; then
A4:(Partial_Sums F).j in Lp_Functions(M,k) &
    F.(j+1) in Lp_Functions(M,k) by A1;
    (Partial_Sums F).(j+1) = (Partial_Sums F).j + F.(j+1) by MESFUN9C:def 2;
    hence P[j+1] by A4,Th25;
   end;
   for j being Nat holds P[j] from NAT_1:sch 2(A2,A3);
   hence thesis;
end;
