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 Th16:
  (for n be Nat holds F.n is E-measurable) implies (
  Partial_Sums F).m is E-measurable
proof
  set PF = Partial_Sums F;
  defpred P[Nat] means PF.$1 is E-measurable;
  assume
A1: for n be Nat holds F.n is E-measurable;
A2: for k being Nat st P[k] holds P[k+1]
  proof
    let k be Nat;
    assume
A3: P[k];
    F.(k+1) is E-measurable by A1;
    then PF.k + F.(k+1) is E-measurable by A3,MESFUNC6:26;
    hence thesis by Def2;
  end;
  PF.0 = F.0 by Def2;
  then
A4: P[ 0 ] by A1;
  for k being Nat holds P[k] from NAT_1:sch 2(A4,A2);
  hence (Partial_Sums F).m is E-measurable;
end;
