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;

theorem Th46:
  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) 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 and
A6: for x be Element of X st x in E holds F#x is convergent;
A7: E = dom lim_sup F by A1,MESFUNC8:def 8;
A8: for x be Element of X, k be Nat st x in E holds -(P.x) <= (F#x).k & (F#x)
  .k <= P.x
  proof
    let x be Element of X, k be Nat;
    assume
A9: x in E;
    then x in dom(F.k) by A1,MESFUNC8:def 2;
    then
A10: x in dom |.(F.k).| by MESFUNC1:def 10;
    (|. F.k .|).x <= P.x by A5,A9;
    then |. (F.k).x .| <= P.x by A10,MESFUNC1:def 10;
    then -(P.x) <= (F.k).x & (F.k).x <= P.x by EXTREAL1:23;
    hence -(P.x) <= (F#x).k & (F#x).k <= P.x by MESFUNC5:def 13;
  end;
A11: now
    let x be Element of X;
    assume
A12: x in dom lim_sup F;
    for k be Nat holds (superior_realsequence(F#x)).k >= -(P.x)
    proof
      let k be Nat;
A13:  (superior_realsequence(F#x)).k >= (F#x).k by RINFSUP2:8;
      (F#x).k >= -(P.x) by A7,A8,A12;
      hence (superior_realsequence(F#x)).k >= -(P.x) by A13,XXREAL_0:2;
    end;
    then lim_sup(F#x) >= -(P.x) by MESFUN10:4;
    then
A14: (lim_sup F).x >= -(P.x) by A12,MESFUNC8:def 8;
    now
      let y be ExtReal;
      assume y in rng(F#x);
      then ex k be object st k in dom(F#x) & y = (F#x).k by FUNCT_1:def 3;
      hence P.x >= y by A7,A8,A12;
    end;
    then P.x is UpperBound of rng(F#x) by XXREAL_2:def 1;
    then P.x >= sup rng(F#x) by XXREAL_2:def 3;
    then P.x >= sup((F#x)^\0) by NAT_1:47;
    then
A15: P.x >= (superior_realsequence(F#x)).0 by RINFSUP2:27;
    (superior_realsequence(F#x)).0 >= inf superior_realsequence(F#x) by
RINFSUP2:23;
    then P.x >= lim_sup(F#x) by A15,XXREAL_0:2;
    then P.x >= (lim_sup F).x by A12,MESFUNC8:def 8;
    hence |. (lim_sup F).x .| <= P.x by A14,EXTREAL1:23;
  end;
A16: now
    let x be Element of X;
    assume
A17: x in dom lim F;
    then x in E by A1,MESFUNC8:def 9;
    then F#x is convergent by A6;
    hence (lim F).x = (lim_sup F).x by A17,MESFUNC8:14;
  end;
A18: lim_sup F is E-measurable by A1,A3,MESFUNC8:23;
  E = dom lim F by A1,MESFUNC8:def 9;
  then lim F = lim_sup F by A7,A16,PARTFUN1:5;
  hence lim F is_integrable_on M by A2,A4,A7,A18,A11,MESFUNC5:102;
end;
