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
  (for m be Nat holds F.m is without+infty) implies (Partial_Sums F).n
  is without+infty
proof
  defpred P[Nat] means (Partial_Sums F).$1 is without+infty;
  assume
A1: for m be Nat holds F.m 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 without+infty by A1;
    (Partial_Sums F).(k+1) = (Partial_Sums F).k + F.(k+1) by Def4;
    hence thesis by A3,A4,Th4;
  end;
  (Partial_Sums F).0 = F.0 by Def4;
  then
A5: P[ 0 ] by A1;
  for k being Nat holds P[k] from NAT_1:sch 2(A5,A2);
  hence thesis;
end;
