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 Th49:
  M.E < +infty & E = dom(F.0) & (for n be Nat holds F.n
is E-measurable) & F is uniformly_bounded & (for x be Element of X st x in E
holds F#x is convergent) implies (for n be Nat holds F.n is_integrable_on M) &
lim F is_integrable_on M & ex I be ExtREAL_sequence st (for n be Nat holds I.n
  = Integral(M,F.n)) & I is convergent & lim I = Integral(M,lim F)
proof
  assume that
A1: M.E < +infty & E = dom(F.0) and
A2: for n be Nat holds F.n is E-measurable and
A3: F is uniformly_bounded and
A4: for x be Element of X st x in E holds F#x is convergent;
  consider K be Real such that
A5: for n be Nat, x be Element of X st x in dom(F.0)
   holds |. (F.n).x qua Complex .|
  <= K by A3;
A6: for x be Element of X st x in E holds (R_EAL F)#x is convergent
  proof
    let x be Element of X;
    assume x in E;
    then
A7: F#x is convergent by A4;
    (R_EAL F)#x = F#x by MESFUN7C:1;
    hence (R_EAL F)#x is convergent by A7,RINFSUP2:14;
  end;
  for n be Nat, x be set st x in dom((R_EAL F).0)
   holds |. ((R_EAL F).n).x.| <= K
  proof
    let n be Nat, x be set;
A8: |. (F.n).x qua Complex .| = |. (F.n).x .| by MESFUNC6:43;
    assume x in dom((R_EAL F).0);
    hence |. ((R_EAL F).n).x .| <= K by A5,A8;
  end;
  then
A9: R_EAL F is uniformly_bounded by MESFUN10:def 1;
A10: for n be Nat holds (R_EAL F).n is E-measurable
  proof
    let n be Nat;
    F.n is E-measurable by A2;
    then R_EAL(F.n) is E-measurable;
    hence (R_EAL F).n is E-measurable;
  end;
  then consider I be ExtREAL_sequence such that
A11: for n be Nat holds I.n = Integral(M,(R_EAL F).n) and
A12: I is convergent & lim I = Integral(M,lim (R_EAL F)) by A1,A9,A6,
MESFUN10:19;
  for n be Nat holds I.n = Integral(M,F.n) by A11;
  hence thesis by A1,A10,A9,A6,A12,MESFUN10:19;
end;
