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 Th47:
  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) & (for x be Element of X st x in E holds
  F#x is convergent) implies lim F is_integrable_on M
proof
  assume that
A1: E = dom(F.0) & E = dom P and
A2: for n be Nat holds F.n is E-measurable and
A3: P is_integrable_on M and
A4: for x be Element of X, n be Nat st x in E holds (|. F.n .|).x <= P.x and
A5: for x be Element of X st x in E holds F#x is convergent;
A6: 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;
A7: 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
A8: F#x is convergent by A5;
    (R_EAL F)#x = F#x by MESFUN7C:1;
    hence (R_EAL F)#x is convergent by A8,RINFSUP2:14;
  end;
A9: 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;
A10: 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 A4,A10;
  end;
  R_EAL P is_integrable_on M by A3;
  hence lim F is_integrable_on M by A1,A6,A9,A7,Th46;
end;
