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 Th51:
  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: for n be Nat holds (Re F).n is E-measurable & (Im F).n
  is E-measurable by A3,Lm2;
  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
A8: |. (Re(F#x)).n qua Complex .|
      <= |. (F#x).n qua Complex .| by COMSEQ_3:13;
    Im((F#x).n1) = (Im(F#x)).n1 by COMSEQ_3:def 6;
    then
A9: |. (Im(F#x)).n qua Complex .|
      <= |. (F#x).n qua Complex .| by COMSEQ_3:13;
    assume
A10: x in E;
    then (|. F.n .|).x <= P.x by A5;
    then |.(F.n).x .| <= P.x by VALUED_1:18;
    then
A11: |. (F#x).n .| <= P.x by MESFUN7C:def 9;
    (Im F)#x = Im(F#x) by A1,A10,MESFUN7C:23;
    then |. ((Im F)#x).n1 qua Complex .| <= P.x by A11,A9,XXREAL_0:2;
    then
A12: |. ((Im F).n1).x qua Complex .| <= P.x by SEQFUNC:def 10;
    (Re F)#x = Re(F#x) by A1,A10,MESFUN7C:23;
    then |. ((Re F)#x).n1 qua Complex .| <= P.x by A11,A8,XXREAL_0:2;
    then |. ((Re F).n1).x qua Complex .| <= P.x by SEQFUNC:def 10;
    hence (|. (Re F).n.|).x <= P.x &
     (|. (Im F).n .|).x <= P.x by A12,
VALUED_1:18;
  end;
  then
A13: for x be Element of X, n be Nat holds (x in E implies (|. (Re F).n .|).x
  <= P.x) & (x in E implies (|. (Im F).n .|).x <= P.x);
A14: for x be Element of X st x in E holds (Re F)#x is convergent & (Im F)#x
  is convergent
  proof
    let x be Element of X;
    assume
A15: x in E;
    then F#x is convergent Complex_Sequence by A6;
    then Re(F#x) is convergent & Im(F#x) is convergent;
    hence (Re F)#x is convergent & (Im F)#x is convergent by A1,A15,MESFUN7C:23
;
  end;
  then
A16: for x be Element of X st x in E holds (Im F)#x is convergent;
  E = dom((Im F).0) by A1,MESFUN7C:def 12;
  then lim Im F is_integrable_on M by A2,A4,A7,A13,A16,Th47;
  then R_EAL(Im lim F) is_integrable_on M by A1,A6,MESFUN7C:25;
  then
A17: Im lim F is_integrable_on M;
A18: for x be Element of X st x in E holds (Re F)#x is convergent by A14;
  E = dom((Re F).0) by A1,MESFUN7C:def 11;
  then lim Re F is_integrable_on M by A2,A4,A7,A13,A18,Th47;
  then R_EAL(Re lim F) is_integrable_on M by A1,A6,MESFUN7C:25;
  then Re lim F is_integrable_on M;
  hence lim F is_integrable_on M by A17,MESFUN6C:def 2;
end;
