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 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
  set PF = Partial_Sums F;
  defpred P1[Nat] means PF.$1 is_integrable_on M;
  assume
A1: for n be Nat holds F.n is_integrable_on M;
A2: for k be Nat st P1[k] holds P1[k+1]
  proof
    let k be Nat;
    assume
A3: P1[k];
    F.(k+1) is_integrable_on M by A1;
    then PF.k + F.(k+1) is_integrable_on M by A3,MESFUNC5:108;
    hence thesis by Def4;
  end;
  PF.0 = F.0 by Def4;
  then
A4: P1[ 0 ] by A1;
  thus for m be Nat holds P1[m] from NAT_1:sch 2(A4,A2);
end;
