reserve n,k for Nat,
  X for non empty set,
  S for SigmaField of X;

theorem Th26:
  for f be with_the_same_dom Functional_Sequence of X,ExtREAL,
      g be PartFunc of X,ExtREAL, E be Element of S st
  dom(f.0) = E &
  (for n be Nat holds f.n is E-measurable) & dom g = E &
  for x be Element of X st x in E holds f#x is convergent &
  g.x = lim(f#x) holds g is E-measurable
proof
  let f be with_the_same_dom Functional_Sequence of X,ExtREAL, g be PartFunc
  of X,ExtREAL, E be Element of S;
  assume that
A1: dom (f.0) = E and
A2: for n be Nat holds f.n is E-measurable and
A3: dom g = E and
A4: for x be Element of X st x in E holds f#x is convergent & g.x = lim( f#x);
A5: dom lim f = E by A1,Def9;
  now
    let x be Element of X;
    assume
A6: x in dom lim f;
    then g.x= lim(f#x) by A4,A5;
    hence g.x = (lim f).x by A6,Def9;
  end;
  then
A7: g = lim f by A3,A5,PARTFUN1:5;
  for x be Element of X st x in E holds f#x is convergent by A4;
  hence thesis by A1,A2,A7,Th25;
end;
