reserve X for non empty set,
  S for SigmaField of X,
  M for sigma_Measure of S,
  E for Element of S,
  F for Functional_Sequence of X,REAL,

  f for PartFunc of X,REAL,
  seq for Real_Sequence,
  n,m for Nat,
  x for Element of X,
  z,D for set;
reserve i for Element of NAT;
reserve F for Functional_Sequence of X,COMPLEX,
  f for PartFunc of X,COMPLEX,
  A for set;
reserve f,g for PartFunc of X,COMPLEX,
  A for Element of S;
reserve F for with_the_same_dom Functional_Sequence of X,ExtREAL,
  P for PartFunc of X,ExtREAL;
reserve F for with_the_same_dom Functional_Sequence of X,REAL,
  f,P for PartFunc of X,REAL;

theorem Th48:
  E = dom(F.0) & E = dom P & (for n be Nat holds F.n
  is E-measurable) & P is_integrable_on M & (for x be Element of X, n be Nat
  st x in E holds (|. F.n .|).x <= P.x) implies ex I be Real_Sequence st (for n
be Nat holds I.n = Integral(M,F.n)) & ( (for x be Element of X st x in E holds
  F#x is convergent) implies I is convergent & lim I = Integral(M,lim F) )
proof
  assume that
A1: E = dom(F.0) and
A2: E = dom P and
A3: for n be Nat holds F.n is E-measurable and
A4: P is_integrable_on M and
A5: for x be Element of X, n be Nat st x in E holds (|. F.n .|).x <= P.x;
A6: R_EAL P is_integrable_on M by A4;
A7: for x be Element of X, n be Nat st x in E holds (|. (R_EAL F).n .|).x <=
  (R_EAL P).x
  proof
    let x be Element of X, n be Nat;
A8: R_EAL |. F.n .| = |. R_EAL (F.n) .| by MESFUNC6:1;
    assume x in E;
    hence (|. (R_EAL F).n .|).x <= (R_EAL P).x by A5,A8;
  end;
A9: for n be Nat holds (R_EAL F).n is E-measurable
  proof
    let n be Nat;
    F.n is E-measurable by A3;
    then R_EAL (F.n) is E-measurable;
    hence (R_EAL F).n is E-measurable;
  end;
  now
    let x be object;
    assume
A10: x in dom P;
    then x in dom|. F.0 .| by A1,A2,VALUED_1:def 11;
    then (|. F.0 .|).x = |. (F.0).x qua Complex .| by VALUED_1:def 11;
    then |. (F.0).x qua Complex .| <= P.x by A2,A5,A10;
    hence 0<= P.x by COMPLEX1:46;
  end;
  then P is nonnegative by MESFUNC6:52;
  then consider J be ExtREAL_sequence such that
A11: for n be Nat holds J.n = Integral(M,(R_EAL F).n) and
  lim_inf J >= Integral(M,lim_inf(R_EAL F)) and
  lim_sup J <= Integral(M,lim_sup(R_EAL F)) and
A12: (for x be Element of X st x in E holds (R_EAL F)#x is convergent)
  implies J is convergent & lim J = Integral(M,lim(R_EAL F)) by A1,A2,A9,A6,A7,
MESFUN10:17;
A13: Integral(M,R_EAL P) < +infty by A6,MESFUNC5:96;
  for n be Nat holds |. J.n .| < +infty
  proof
    let n be Nat;
A14: E = dom((R_EAL F).n) & (R_EAL F).n is E-measurable by A1,A9,
MESFUNC8:def 2;
    |. (R_EAL F).n .| is_integrable_on M by A1,A2,A9,A6,A7,MESFUN10:16;
    then (R_EAL F).n is_integrable_on M by A14,MESFUNC5:100;
    then
A15: |. Integral(M,(R_EAL F).n) .| <= Integral(M,|. (R_EAL F).n .|) by
MESFUNC5:101;
    for x be Element of X st x in dom((R_EAL F).n) holds |. ((R_EAL F).n)
    .x .| <= (R_EAL P).x
    proof
      let x be Element of X;
      assume
A16:  x in dom((R_EAL F).n);
      then x in E by A1,MESFUNC8:def 2;
      then
A17:  (|. (R_EAL F).n .|).x <= (R_EAL P).x by A7;
      x in dom |. (R_EAL F).n .| by A16,MESFUNC1:def 10;
      hence |. ((R_EAL F).n).x .| <= (R_EAL P).x by A17,MESFUNC1:def 10;
    end;
    then Integral(M,|. (R_EAL F).n .|) <= Integral(M,R_EAL P) by A2,A6,A14,
MESFUNC5:102;
    then |. Integral(M,(R_EAL F).n) .| <= Integral(M,R_EAL P) by A15,XXREAL_0:2
;
    then |. Integral(M,(R_EAL F).n) .| < +infty by A13,XXREAL_0:2;
    hence |. J.n .| < +infty by A11;
  end;
  then for n be Element of NAT st n in dom J holds |. J.n .| < +infty;
  then J is real-valued by MESFUNC2:def 1;
  then reconsider I=J as Real_Sequence by RINFSUP2:6;
  (for x be Element of X st x in E holds F#x is convergent) implies J is
  convergent_to_finite_number & lim J = Integral(M,lim F)
  proof
    assume
A18: for x be Element of X st x in E holds F#x is convergent;
A19: now
      let x be Element of X;
      assume x in E;
      then
A20:  F#x is convergent by A18;
      F#x = (R_EAL F)#x by MESFUN7C:1;
      hence (R_EAL F)#x is convergent by A20,RINFSUP2:14;
    end;
    lim F is_integrable_on M by A1,A2,A3,A4,A5,A18,Th47;
    then
A21: -infty < Integral(M,lim F) & Integral(M,lim F) < +infty by MESFUNC5:96;
    J is convergent implies J is convergent_to_finite_number
    by A12,A19,A21,MESFUNC5:def 12;
    hence thesis by A12,A19;
  end;
  then
A22: (for x be Element of X st x in E holds F#x is convergent) implies I is
  convergent & lim I = Integral(M,lim F) by RINFSUP2:15;
  for n be Nat holds I.n = Integral(M,F.n) by A11;
  hence thesis by A22;
end;
