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 Th41:
  (for n be Nat holds F.n is E-measurable & F.n is
  without-infty) 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 & F.n is without-infty;
A2: for k being Nat st P[k] holds P[k+1]
  proof
    let k be Nat;
    assume
A3: P[k];
A4: F.(k+1) is E-measurable by A1;
A5: F.(k+1) is without-infty by A1;
    PF.k is without-infty by A1,Th39;
    then PF.k + F.(k+1) is E-measurable by A3,A4,A5,MESFUNC5:31;
    hence thesis by Def4;
  end;
  PF.0 = F.0 by Def4;
  then
A6: P[ 0 ] by A1;
  for k being Nat holds P[k] from NAT_1:sch 2(A6,A2);
  hence thesis;
end;
