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;

theorem
  dom(F.0) = E & F is with_the_same_dom & (for n be Nat holds (
Partial_Sums F).n is E-measurable) & (for x be Element of X st x in E holds
  F#x is summable) implies lim(Partial_Sums F) is E-measurable
proof
  assume that
A1: dom(F.0) = E and
A2: F is with_the_same_dom and
A3: for n be Nat holds (Partial_Sums F).n is E-measurable and
A4: for x be Element of X st x in E holds F#x is summable;
A5: dom((Im F).0) = E by A1,MESFUN7C:def 12;
A6: for x be Element of X st x in dom((Partial_Sums F).0) holds (
  Partial_Sums F)#x is convergent
  proof
    let x be Element of X;
    assume
A7: x in dom((Partial_Sums F).0);
A8: dom((Partial_Sums F).0) = dom(F.0) by A2,Th32;
    then F#x is summable by A1,A4,A7;
    then Partial_Sums (F#x) is convergent;
    hence (Partial_Sums F)#x is convergent by A2,A7,A8,Th35;
  end;
A9: for n be Nat holds (Partial_Sums Im F).n is E-measurable
  proof
    let n be Nat;
    (Partial_Sums F).n is E-measurable by A3;
    then Im((Partial_Sums F).n) is E-measurable by MESFUN6C:def 1;
    then (Im(Partial_Sums F)).n is E-measurable by MESFUN7C:24;
    hence (Partial_Sums Im F).n is E-measurable by Th29;
  end;
A10: for x be Element of X st x in E holds (Im F)#x is summable
  proof
    let x be Element of X;
    assume
A11: x in E;
    then F#x is summable by A4;
    then Im(F#x) is summable;
    hence (Im F)#x is summable by A1,A2,A11,MESFUN7C:23;
  end;
A12: Re F is with_the_same_dom by A2;
  then Im F is with_the_same_dom by Th25;
  then lim(Partial_Sums Im F) is E-measurable by A5,A9,A10,Th18;
  then
A13: lim(Im(Partial_Sums F)) is E-measurable by Th29;
A14: for x be Element of X st x in E holds (Re F)#x is summable
  proof
    let x be Element of X;
    assume
A15: x in E;
    then F#x is summable by A4;
    then Re(F#x) is summable;
    hence (Re F)#x is summable by A1,A2,A15,MESFUN7C:23;
  end;
A16: for n be Nat holds (Partial_Sums Re F).n is E-measurable
  proof
    let n be Nat;
    (Partial_Sums F).n is E-measurable by A3;
    then Re((Partial_Sums F).n) is E-measurable by MESFUN6C:def 1;
    then (Re(Partial_Sums F)).n is E-measurable by MESFUN7C:24;
    hence (Partial_Sums Re F).n is E-measurable by Th29;
  end;
A17: Partial_Sums F is with_the_same_dom by A2,Th34;
  then lim(Im(Partial_Sums F)) = R_EAL Im(lim(Partial_Sums F)) by A6,
MESFUN7C:25;
  then
A18: Im(lim(Partial_Sums F)) is E-measurable by A13;
  dom((Re F).0) = E by A1,MESFUN7C:def 11;
  then lim(Partial_Sums Re F) is E-measurable by A12,A16,A14,Th18;
  then
A19: lim(Re(Partial_Sums F)) is E-measurable by Th29;
  lim(Re(Partial_Sums F)) = R_EAL Re(lim(Partial_Sums F)) by A6,A17,MESFUN7C:25
;
  then Re(lim(Partial_Sums F)) is E-measurable by A19;
  hence lim(Partial_Sums F) is E-measurable by A18,MESFUN6C:def 1;
end;
