reserve X for non empty set,
  S for SigmaField of X,
  M for sigma_Measure of S,
  E for Element of S,
  F for Functional_Sequence of X,REAL,

  f for PartFunc of X,REAL,
  seq for Real_Sequence,
  n,m for Nat,
  x for Element of X,
  z,D for set;
reserve i for Element of NAT;

theorem Th19:
  (for n be Nat holds F.n is_integrable_on M) implies for m be Nat
  holds (Partial_Sums F).m is_integrable_on M
proof
  assume
A1: for n be Nat holds F.n is_integrable_on M;
  let m be Nat;
  for n be Nat holds (R_EAL F).n is_integrable_on M
  proof
    let n be Nat;
    F.n is_integrable_on M by A1;
    then R_EAL(F.n) is_integrable_on M;
    hence (R_EAL F).n is_integrable_on M;
  end;
  then (Partial_Sums R_EAL F).m is_integrable_on M by MESFUNC9:45;
  then ((R_EAL(Partial_Sums F)).m) is_integrable_on M by Th7;
  then (R_EAL((Partial_Sums F).m)) is_integrable_on M;
  hence (Partial_Sums F).m is_integrable_on M;
end;
