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 Th50:
  E = dom(F.0) & F is additive & F is with_the_same_dom & (for n
  be Nat holds F.n is E-measurable & F.n is nonnegative) implies ex I be
  ExtREAL_sequence st for n be Nat holds I.n = Integral(M,F.n) & Integral(M,(
  Partial_Sums F).n) = (Partial_Sums I).n
proof
  assume that
A1: E = dom(F.0) and
A2: F is additive and
A3: F is with_the_same_dom and
A4: for n be Nat holds F.n is E-measurable & F.n is nonnegative;
  set PF = Partial_Sums F;
  deffunc I(Element of NAT) = Integral(M,F.$1);
  consider I be sequence of ExtREAL such that
A5: for n be Element of NAT holds I.n = I(n) from FUNCT_2:sch 4;
  reconsider I as ExtREAL_sequence;
  take I;
A6: for n be Nat holds F.n is without-infty by A4,MESFUNC5:12;
  thus for n be Nat holds I.n = Integral(M,F.n) & Integral(M,(Partial_Sums F).
  n) = (Partial_Sums I).n
  proof
    set PI = Partial_Sums I;
    defpred P2[Nat] means Integral(M,PF.$1) = (Partial_Sums I).$1;
    let n be Nat;
    reconsider n9=n as Element of NAT by ORDINAL1:def 12;
    I.n = Integral(M,F.n9) by A5;
    hence I.n = Integral(M,F.n);
A7: for k be Nat st P2[k] holds P2[k+1]
    proof
      let k be Nat;
      assume
A8:   P2[k];
A9:   F.(k+1) is E-measurable by A4;
A10:  dom(F.(k+1)) = E by A1,A3;
A11:  PF.(k+1) is E-measurable by A4,A6,Th41;
A12:  F.(k+1) is nonnegative by A4;
A13:  PF.k is nonnegative by A4,Th36;
A14:  dom(PF.k) = E by A1,A2,A3,Th29;
A15:  PF.k is E-measurable by A4,A6,Th41;
      then consider D be Element of S such that
A16:  D = dom(PF.k + F.(k+1)) and
A17:  integral+(M,PF.k + F.(k+1)) = integral+(M,(PF.k)|D) + integral+
      (M,( F.(k+1))|D) by A14,A10,A9,A13,A12,MESFUNC5:78;
A18:  D = dom(PF.k) /\ dom(F.(k+1)) by A13,A12,A16,MESFUNC5:22
        .= E by A14,A10;
      then
A19:  (PF.k)|D = PF.k by A14;
A20:  (F.(k+1))|D = F.(k+1) by A10,A18;
      dom(PF.(k+1)) = E by A1,A2,A3,Th29;
      then Integral(M,PF.(k+1)) = integral+(M,PF.(k+1)) by A4,A11,Th36,
MESFUNC5:88
        .= integral+(M,(PF.k)|D) + integral+(M,(F.(k+1))|D) by A17,Def4
        .= Integral(M,PF.k) + integral+(M,(F.(k+1))|D) by A14,A15,A13,A19,
MESFUNC5:88
        .= Integral(M,PF.k) + Integral(M,F.(k+1)) by A10,A9,A12,A20,MESFUNC5:88
        .= PI.k + I.(k+1) by A5,A8;
      hence thesis by Def1;
    end;
    Integral(M,PF.0) = Integral(M,F.0) by Def4;
    then Integral(M,PF.0) = I.0 by A5;
    then
A21: P2[ 0 ] by Def1;
    for k be Nat holds P2[k] from NAT_1:sch 2(A21,A7);
    hence thesis;
  end;
end;
