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;
reserve F for with_the_same_dom Functional_Sequence of X,COMPLEX,
  f for PartFunc of X,COMPLEX;

theorem
  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 Complex_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 and
A2: E = dom(F.0) and
A3: for n be Nat holds F.n is E-measurable and
A4: F is uniformly_bounded and
A5: for x be Element of X st x in E holds F#x is convergent;
  consider K be Real such that
A6: for n be Nat, x be Element of X st x in dom(F.0) holds |. (F.n).x .|
  <= K by A4;
A7: for x be Element of X st x in E holds (Re F)#x is convergent
  proof
    let x be Element of X;
    assume
A8: x in E;
    then F#x is convergent Complex_Sequence by A5;
    then Re(F#x) is convergent;
    hence (Re F)#x is convergent by A2,A8,MESFUN7C:23;
  end;
  for n be Nat, x be Element of X st x in dom((Im F).0)
    holds |. ((Im F). n).x qua Complex .| <= K
  proof
    let n be Nat, x be Element of X;
    assume x in dom((Im F).0);
    then
A9: x in E by A2,MESFUN7C:def 12;
    then |. (F.n).x .| <= K by A2,A6;
    then
A10: |. (F#x).n .| <= K by MESFUN7C:def 9;
    Im((F#x).n) = (Im(F#x)).n by COMSEQ_3:def 6;
    then Im((F#x).n) = ((Im F)#x).n by A2,A9,MESFUN7C:23;
    then |. ((Im F)#x).n qua Complex .| <= |. (F#x).n .| by COMSEQ_3:13;
    then |. ((Im F)#x).n qua Complex .| <= K by A10,XXREAL_0:2;
    hence thesis by SEQFUNC:def 10;
  end;
  then
A11: Im F is uniformly_bounded;
A12: for x be Element of X st x in E holds (Im F)#x is convergent
  proof
    let x be Element of X;
    assume
A13: x in E;
    then F#x is convergent Complex_Sequence by A5;
    then Im(F#x) is convergent;
    hence (Im F)#x is convergent by A2,A13,MESFUN7C:23;
  end;
  defpred P[Element of NAT,set] means $2 = Integral(M,F.$1);
A14: for n being Element of NAT ex y being Element of COMPLEX st P[n,y]
  proof
    let n being Element of NAT;
    take Integral(M,F.n);
    thus thesis by XCMPLX_0:def 2,TARSKI:1;
  end;
  consider I be sequence of COMPLEX such that
A15: for n be Element of NAT holds P[n,I.n] from FUNCT_2:sch 3(A14);
  reconsider I as Complex_Sequence;
A16: for n be Nat holds (Re F).n is E-measurable & (Im F).n
  is E-measurable by A3,Lm2;
  for n be Nat, x be Element of X st x in dom((Re F).0)
     holds |. ((Re F).n).x qua Complex .| <= K
  proof
    let n be Nat, x be Element of X;
    assume x in dom((Re F).0);
    then
A17: x in E by A2,MESFUN7C:def 11;
    then |. (F.n).x .| <= K by A2,A6;
    then
A18: |. (F#x).n .| <= K by MESFUN7C:def 9;
    Re((F#x).n) = (Re(F#x)).n by COMSEQ_3:def 5;
    then Re((F#x).n) = ((Re F)#x).n by A2,A17,MESFUN7C:23;
    then |. ((Re F)#x).n qua Complex .| <= |. (F#x).n .| by COMSEQ_3:13;
    then |. ((Re F)#x).n qua Complex .| <= K by A18,XXREAL_0:2;
    hence |. ((Re F).n).x qua Complex .| <= K by SEQFUNC:def 10;
  end;
  then
A19: Re F is uniformly_bounded;
A20: E = dom((Im F).0) by A2,MESFUN7C:def 12;
  then consider B be ExtREAL_sequence such that
A21: for n be Nat holds B.n = Integral(M,(Im F).n) and
A22: B is convergent and
A23: lim B = Integral(M,lim Im F) by A1,A16,A12,A11,Th49;
A24: lim Im F is_integrable_on M by A1,A20,A16,A12,A11,Th49;
  then R_EAL Im lim F is_integrable_on M by A2,A5,MESFUN7C:25;
  then
A25: Im lim F is_integrable_on M;
A26: E = dom((Re F).0) by A2,MESFUN7C:def 11;
  then consider A be ExtREAL_sequence such that
A27: for n be Nat holds A.n = Integral(M,(Re F).n) and
A28: A is convergent and
A29: lim A = Integral(M,lim Re F) by A1,A16,A7,A19,Th49;
  thus
A30: for n be Nat holds F.n is_integrable_on M
  proof
    let n be Nat;
    (Im F).n = Im(F.n) by MESFUN7C:24;
    then
A31: Im(F.n) is_integrable_on M by A1,A20,A16,A12,A11,Th49;
    (Re F).n = Re(F.n) by MESFUN7C:24;
    then Re(F.n) is_integrable_on M by A1,A26,A16,A7,A19,Th49;
    hence F.n is_integrable_on M by A31,MESFUN6C:def 2;
  end;
A32: for n1 be set st n1 in NAT holds (R_EAL Re I).n1 = A.n1 & (R_EAL Im I).
  n1 = B.n1
  proof
    let n1 be set;
    assume n1 in NAT;
    then reconsider n=n1 as Element of NAT;
A33: I.n1 = Integral(M,F.n) by A15;
    F.n is_integrable_on M by A30;
    then consider RF,IF be Real such that
A34: RF = Integral(M,Re(F.n)) and
A35: IF = Integral(M,Im(F.n)) and
A36: Integral(M,F.n) = RF + IF * <i> by MESFUN6C:def 3;
    (Re I).n = Re(I.n) by COMSEQ_3:def 5;
    then (Re F).n = Re(F.n) & (Re I).n1 = RF by A36,A33,COMPLEX1:12,MESFUN7C:24
;
    hence (R_EAL Re I).n1 = A.n1 by A27,A34;
    (Im I).n = Im(I.n) by COMSEQ_3:def 6;
    then (Im F).n = Im(F.n) & (Im I).n1 = IF by A36,A33,COMPLEX1:12,MESFUN7C:24
;
    hence (R_EAL Im I).n1 = B.n1 by A21,A35;
  end;
  then for n1 be object st n1 in NAT holds (R_EAL Im I).n1 = B.n1;
  then
A37: Im I = B;
A38: -infty < Integral(M,lim Im F) & Integral(M,lim Im F) < +infty by A24,
MESFUNC5:96;
A39: B is convergent implies B is convergent_to_finite_number
     by A23,A38,MESFUNC5:def 12;
  then
A40: lim Im I = Integral(M,lim Im F) by A22,A23,A37,RINFSUP2:15;
A41: Im I is convergent by A22,A37,A39,RINFSUP2:15;
A42: lim Re F is_integrable_on M by A1,A26,A16,A7,A19,Th49;
  then R_EAL Re lim F is_integrable_on M by A2,A5,MESFUN7C:25;
  then Re lim F is_integrable_on M;
  hence
A43: lim F is_integrable_on M by A25,MESFUN6C:def 2;
  take I;
  for n1 be object st n1 in NAT holds (R_EAL Re I).n1 = A.n1 by A32;
  then
A44: Re I = A;
  thus for n be Nat holds I.n = Integral(M,F.n)
  proof
    let n be Nat;
A45: (Re I).n = Re(I.n) & (Im I).n = Im(I.n) by COMSEQ_3:def 5,def 6;
    (Re F).n = Re(F.n) by MESFUN7C:24;
    then
A46: (Re I).n = Integral(M,Re(F.n)) by A27,A44;
    (Im F).n = Im(F.n) by MESFUN7C:24;
    then
A47: (Im I).n = Integral(M,Im(F.n)) by A21,A37;
    I.n = Re(I.n) + Im(I.n)*<i> & F.n is_integrable_on M by A30,COMPLEX1:13;
    hence I.n = Integral(M,F.n) by A45,A46,A47,MESFUN6C:def 3;
  end;
A48: -infty < Integral(M,lim Re F) & Integral(M,lim Re F) < +infty by A42,
MESFUNC5:96;
A49: A is convergent implies A is convergent_to_finite_number
     by A29,A48,MESFUNC5:def 12;
  then
A50: Re I is convergent by A28,A44,RINFSUP2:15;
  hence I is convergent by A41,COMSEQ_3:42;
  for n be Nat holds (Re I).n = Re(I.n) & (Im I).n = Im(I.n)
  by COMSEQ_3:def 5,def 6;
  then
A51: lim I = lim Re I + (lim Im I) * <i> by A50,A41,COMSEQ_3:39;
A52: Integral(M,lim Re F) = Integral(M,Re lim F) & Integral(M,lim Im F) =
  Integral(M,Im lim F) by A2,A5,MESFUN7C:25;
  lim Re I = Integral(M,lim Re F) by A28,A29,A44,A49,RINFSUP2:15;
  hence lim I = Integral(M,lim F) by A43,A40,A51,A52,MESFUN6C:def 3;
end;
