reserve X for non empty set,
  Y for set,
  S for SigmaField of X,
  M for sigma_Measure of S,
  f,g for PartFunc of X,COMPLEX,
  r for Real,
  k for Real,
  n for Nat,
  E for Element of S;

theorem
  f is_integrable_on M implies ex F be sequence of S st (for n be
Nat holds F.n = dom f /\ great_eq_dom(|.f.|, (1/(n+1)))) & dom f \ eq_dom(
  |.f.|,0) = union rng F & for n be Nat holds F.n in S & M.(F.n) <+infty
proof
  assume
A1: f is_integrable_on M;
  then consider E be Element of S such that
A2: E = dom f and
A3: f is E-measurable by Th35;
  defpred P[Element of NAT,set] means $2 = E /\ great_eq_dom(|.f.|,1/($1+1));
A4: dom |.f.| = E by A2,VALUED_1:def 11;
  now
    let x be object;
    reconsider y=|.f.|.x as Real;
    assume
A5: x in E \ eq_dom(|.f.|, 0);
    then
A6: x in E by XBOOLE_0:def 5;
    then
A7: x in dom |.f.| by A2,VALUED_1:def 11;
    not x in eq_dom(|.f.|, 0) by A5,XBOOLE_0:def 5;
    then not y = 0 by A7,MESFUNC6:7;
    then not |.f.x.| = 0 by A7,VALUED_1:def 11;
    then f.x <> 0 by COMPLEX1:5,SQUARE_1:17;
    then 0 < |.f.x.| by COMPLEX1:47;
    then 0 < (|.f.|).x by A7,VALUED_1:def 11;
    then x in great_dom(|.f.|, 0) by A7,MESFUNC1:def 13;
    hence x in E /\ great_dom(|.f.|, 0) by A6,XBOOLE_0:def 4;
  end;
  then
A8: E \ eq_dom(|.f.|, 0) c= E /\ great_dom(|.f.|, 0);
  now
    let x be object;
    assume
A9: x in E /\ great_dom(|.f.|, 0);
    then x in great_dom(|.f.|, 0) by XBOOLE_0:def 4;
    then 0 < (|.f.|).x by MESFUNC1:def 13;
    then
A10: not x in eq_dom(|.f.|, 0) by MESFUNC1:def 15;
    x in E by A9,XBOOLE_0:def 4;
    hence x in E \ eq_dom(|.f.|, 0) by A10,XBOOLE_0:def 5;
  end;
  then
A11: E /\ great_dom(|.f.|, 0) c= E \ eq_dom(|.f.|, 0);
A12: |.f.| is E-measurable by A2,A3,MESFUN6C:30;
A13: for n be Element of NAT ex Z be Element of S st P[n,Z]
  proof
    let n be Element of NAT;
    take E /\ great_eq_dom(|.f.|,1/(n+1));
    thus thesis by A12,A4,MESFUNC6:13;
  end;
  consider F be sequence of S such that
A14: for n be Element of NAT holds P[n,F.n] from FUNCT_2:sch 3(A13);
A15: for n be Nat holds F.n in S & M.(F.n) <+infty
  proof
    |.f.| is_integrable_on M by A1,Th35;
    then
A16: Integral(M,|.f.|) < +infty by MESFUNC6:90;
    let n be Nat;
    reconsider n1=n as Element of NAT by ORDINAL1:def 12;
    set z = (1/(n+1));
A17: F.n1 = E /\ great_eq_dom(|.f.| ,1/(n1+1)) by A14;
    then reconsider En=F.n as Element of S;
    set h = (|.f.|)|En;
    consider nf be PartFunc of X,REAL such that
A18: nf is_simple_func_in S and
A19: dom nf = En and
A20: for x be object st x in En holds nf.x = 1/(n+1) by MESFUNC6:75;
A21: dom h = En by A4,A17,RELAT_1:62,XBOOLE_1:17;
A22: F.n c= great_eq_dom(|.f.|, 1/(n+1)) by A17,XBOOLE_1:17;
A23: for x be Element of X st x in dom nf holds nf.x <= h.x
    proof
      let x be Element of X;
      assume
A24:  x in dom nf;
      then h.x = |.f.| .x by A19,FUNCT_1:49;
      then 1/(n+1) <= h.x by A22,A19,A24,MESFUNC1:def 14;
      hence thesis by A19,A20,A24;
    end;
    dom |.f.| /\ En = E /\ En by A2,VALUED_1:def 11;
    then
A25: dom |.f.| /\ En = En by A17,XBOOLE_1:17,28;
    |.f.| is En-measurable by A12,A17,MESFUNC6:16,XBOOLE_1:17;
    then
A26: h is En-measurable by A25,MESFUNC6:76;
A27: h is nonnegative by Lm1,MESFUNC6:55;
    for x be object st x in dom nf holds nf.x >= 0 by A19,A20;
    then
A28: nf is nonnegative by MESFUNC6:52;
    |.f.| is nonnegative & (|.f.|)|E=|.f.| by A4,Lm1;
    then
A29: Integral(M,h) <= Integral(M,|.f.|) by A12,A4,A17,MESFUNC6:87,XBOOLE_1:17;
    nf is En-measurable by A18,MESFUNC6:50;
    then Integral(M,nf) <= Integral(M,h) by A21,A26,A27,A19,A28,A23,Th34;
    then
A30: Integral(M,nf) <= Integral(M,|.f.|) by A29,XXREAL_0:2;
A31: z* M.En / z = M.En & +infty / z = +infty by XXREAL_3:83,88;
    Integral(M,nf) = (1/(n+1)) * M.En by A19,A20,MESFUNC6:97;
    then (1/(n+1)) * M.En < +infty by A16,A30,XXREAL_0:2;
    hence thesis by A31,XXREAL_3:80;
  end;
  take F;
A32: for n be Nat holds F.n = E /\ great_eq_dom(|.f.|, 1/(n+1))
  proof
    let n be Nat;
    n in NAT by ORDINAL1:def 12;
    hence thesis by A14;
  end;
  then for n be Nat holds F.n = E /\ great_eq_dom(|.f.|, 0 + 1/(n+1));
  then E /\ great_dom(|.f.|, 0) = union rng F by MESFUNC6:11;
  hence thesis by A2,A32,A11,A8,A15;
end;
