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
  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) implies ex I be Complex_Sequence st (for n be Nat holds I.n =
Integral(M,F.n)) & ( (for x be Element of X st x in E holds F#x is convergent)
  implies I is convergent & lim I = Integral(M,lim F) )
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;
A6: E = dom((Re F).0) by A1,MESFUN7C:def 11;
A7: for n be Nat holds (Re F).n is E-measurable & (Im F).n
  is E-measurable by A3,Lm2;
A8: for x be Element of X, n be Nat st x in E holds (|. (Re F).n .|).x <= P.
  x & (|. (Im F).n .|).x <= P.x
  proof
    let x be Element of X, n be Nat;
    reconsider n1=n as Element of NAT by ORDINAL1:def 12;
    Re((F#x).n1) = (Re(F#x)).n1 by COMSEQ_3:def 5;
    then
A9: |. (Re(F#x)).n qua Complex .| <= |. (F#x).n .| by COMSEQ_3:13;
    Im((F#x).n1) = (Im(F#x)).n1 by COMSEQ_3:def 6;
    then
A10: |. (Im(F#x)).n qua Complex .| <= |. (F#x).n .| by COMSEQ_3:13;
    assume
A11: x in E;
    then (|. F.n .|).x <= P.x by A5;
    then |.(F.n).x .| <= P.x by VALUED_1:18;
    then
A12: |. (F#x).n .| <= P.x by MESFUN7C:def 9;
    (Im F)#x = Im(F#x) by A1,A11,MESFUN7C:23;
    then
A13: |. ((Im F)#x).n1 qua Complex .| <= P.x by A12,A10,XXREAL_0:2;
    (Re F)#x = Re(F#x) by A1,A11,MESFUN7C:23;
    then |. ((Re F)#x).n qua Complex .| <= P.x by A12,A9,XXREAL_0:2;
    then
A14: |. ((Re F).n).x qua Complex .| <= P.x by SEQFUNC:def 10;
    |. ((Im F).n).x qua Complex .| <= P.x by A13,SEQFUNC:def 10;
    hence (|. (Re F).n .|).x <= P.x &
     (|. (Im F).n .|).x <= P.x by A14,
VALUED_1:18;
  end;
  then
  for x be Element of X, n be Nat st x in E holds (|. (Re F).n .|).x <= P .x;
  then consider A be Real_Sequence such that
A15: for n be Nat holds A.n = Integral(M,(Re F).n) and
A16: (for x be Element of X st x in E holds (Re F)#x is convergent)
  implies A is convergent & lim A = Integral(M,lim Re F) by A2,A4,A6,A7,Th48;
  defpred P[Element of NAT,set] means $2 = Integral(M,F.$1);
A17: for n being Element of NAT ex y being Element of COMPLEX st P[n,y]
  proof
    let n be Element of NAT;
    Integral(M,F.n) is Element of COMPLEX by XCMPLX_0:def 2;
    hence thesis;
  end;
  consider I be sequence of COMPLEX such that
A18: for n be Element of NAT holds P[n,I.n] from FUNCT_2:sch 3(A17);
  reconsider I as Complex_Sequence;
A19: E = dom((Im F).0) by A1,MESFUN7C:def 12;
  for x be Element of X, n be Nat st x in E holds (|. (Im F).n .|).x <= P
  .x by A8;
  then consider B be Real_Sequence such that
A20: for n be Nat holds B.n = Integral(M,(Im F).n) and
A21: (for x be Element of X st x in E holds (Im F)#x is convergent)
  implies B is convergent & lim B = Integral(M,lim Im F) by A2,A4,A19,A7,Th48;
A22: for n be Nat holds (Re F).n is_integrable_on M & (Im F).n
  is_integrable_on M
  proof
    let n be Nat;
A23: now
      let x be Element of X;
      assume x in dom((Re F).n);
      then x in E by A6,MESFUNC8:def 2;
      then (|. (Re F).n .|).x <= P.x by A8;
      hence |. ((Re F).n).x qua Complex .| <= P.x by VALUED_1:18;
    end;
A24: now
      let x be Element of X;
      assume x in dom((Im F).n);
      then x in E by A19,MESFUNC8:def 2;
      then (|. (Im F).n .|).x <= P.x by A8;
      hence |. ((Im F).n).x qua Complex .| <= P.x by VALUED_1:18;
    end;
A25: (Re F).n is E-measurable & (Im F).n is E-measurable by A3,Lm2;
    dom((Re F).n) = E & dom((Im F).n) = E by A6,A19,MESFUNC8:def 2;
    hence (Re F).n is_integrable_on M & (Im F).n is_integrable_on M by A2,A4
,A23,A24,A25,MESFUNC6:96;
  end;
A26: now
    let n1 be set;
    assume n1 in NAT;
    then reconsider n=n1 as Element of NAT;
A27: (Re I).n = Re(I.n) & (Im I).n = Im(I.n) by COMSEQ_3:def 5,def 6;
A28: (Im F).n = Im(F.n) by MESFUN7C:24;
    then
A29: Im(F.n) is_integrable_on M by A22;
A30: (Re F).n = Re(F.n) by MESFUN7C:24;
    then Re(F.n) is_integrable_on M by A22;
    then F.n is_integrable_on M by A29,MESFUN6C:def 2;
    then consider RF,IF be Real such that
A31: RF = Integral(M,Re(F.n)) & IF = Integral(M,Im(F.n)) and
A32: Integral(M,F.n) = RF + IF * <i> by MESFUN6C:def 3;
    I.n1 = Integral(M,F.n) by A18;
    then Re(I.n1) = RF & Im(I.n1) = IF by A32,COMPLEX1:12;
    hence (Re I).n1 = A.n1 & (Im I).n1 = B.n1 by A15,A20,A30,A28,A31,A27;
  end;
  then for x be object st x in NAT holds (Re I).x = A.x;
  then
A33: Re I = A;
  take I;
  hereby
    let n be Nat;
    n is Element of NAT by ORDINAL1:def 12;
    hence I.n = Integral(M,F.n) by A18;
  end;
  for x be object st x in NAT holds (Im I).x = B.x by A26;
  then
A34: Im I = B;
  hereby
    assume
A35: for x be Element of X st x in E holds F#x is convergent;
    then
A36: Integral(M,lim Im F) = Integral(M,Im lim F) by A1,MESFUN7C:25;
A37: lim F is_integrable_on M & Integral(M,lim Re F) = Integral(M,Re lim F
    ) by A1,A2,A3,A4,A5,A35,Th51,MESFUN7C:25;
A38: now
      let x be Element of X such that
A39:  x in E;
      F#x is convergent Complex_Sequence by A35,A39;
      then Re(F#x) is convergent & Im(F#x) is convergent;
      hence (Re F)#x is convergent & (Im F)#x is convergent by A1,A39,
MESFUN7C:23;
    end;
    hence I is convergent by A16,A21,A33,A34,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 lim I = lim Re I + (lim Im I) * <i> by A16,A21,A33,A34,A38,COMSEQ_3:39
;
    hence lim I = Integral(M,lim F) by A16,A21,A33,A34,A38,A37,A36,
MESFUN6C:def 3;
  end;
end;
