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 Th50:
  M.E < +infty & E = dom(F.0) & (for n be Nat holds F.n
is_integrable_on M) & F is_uniformly_convergent_to f implies 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,f)
proof
  assume that
A1: M.E < +infty & E = dom(F.0) and
A2: for n be Nat holds F.n is_integrable_on M and
A3: F is_uniformly_convergent_to f;
A4: for n be Nat holds (R_EAL F).n is_integrable_on M
  proof
    let n be Nat;
    F.n is_integrable_on M by A2;
    then R_EAL(F.n) is_integrable_on M;
    hence (R_EAL F).n is_integrable_on M;
  end;
A5: for e be Real st e>0 ex N be Nat st for n be Nat, x be set st n
  >= N & x in dom((R_EAL F).0) holds |. ((R_EAL F).n).x - (R_EAL f).x .| < e
  proof
    let e be Real;
    assume e>0;
    then consider N be Nat such that
A6: for n be Nat, x be Element of X st n >= N & x in dom(F.0) holds
    |. (F.n).x - f.x qua Complex .| < e by A3;
    now
      let n be Nat, x be set;
      assume n >= N & x in dom((R_EAL F).0);
      then
A7:   |. (F.n).x - f.x qua Complex .| < e by A6;
      |. (F.n).x - f.x qua Complex .|
        = |. ((F.n).x - f.x) .| by MESFUNC6:43;
      hence |. ((R_EAL F).n).x - (R_EAL f).x .| < e by A7;
    end;
    hence thesis;
  end;
  dom((R_EAL F).0) = dom R_EAL f by A3;
  then
A8: R_EAL F is_uniformly_convergent_to R_EAL f by A5,MESFUN10:def 2;
  then
A9: R_EAL f is_integrable_on M by A1,A4,MESFUN10:21;
  consider I be ExtREAL_sequence such that
A10: for n be Nat holds I.n = Integral(M,(R_EAL F).n) and
A11: I is convergent & lim I = Integral(M,R_EAL f) by A1,A4,A8,MESFUN10:21;
  for n be Nat holds I.n = Integral(M,F.n) by A10;
  hence thesis by A9,A11;
end;
