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;

theorem Th18:
  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 & F is with_the_same_dom and
A2: for n be Nat holds (Partial_Sums F).n is E-measurable and
A3: for x be Element of X st x in E holds F#x is summable;
A4: now
    let x be Element of X;
    assume
A5: x in E;
    then F#x is summable by A3;
    then Partial_Sums(F#x) is convergent;
    hence (Partial_Sums F)#x is convergent by A1,A5,Th13;
  end;
  dom((Partial_Sums F).0) = E & Partial_Sums F is with_the_same_dom
  Functional_Sequence of X,REAL by A1,Th11,Th17;
  hence thesis by A2,A4,MESFUN7C:21;
end;
