reserve X for non empty set,
  Y for set,
  S for SigmaField of X,
  M for sigma_Measure of S,
  f,g for PartFunc of X,COMPLEX,
  r for Real,
  k for Real,
  n for Nat,
  E for Element of S;

theorem Th22:
  for f be with_the_same_dom Functional_Sequence of X,REAL, 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,REAL, 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,MESFUNC8:def 9;
  now
    let x be Element of X;
    assume
A6: x in dom lim f;
    then x in E by A1,MESFUNC8:def 9;
    then f#x is convergent by A4;
    then lim(f#x) = lim R_EAL(f#x) by RINFSUP2:14;
    then g.x = lim R_EAL(f#x) by A4,A5,A6;
    hence g.x = (lim f).x by A6,Th14;
  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,Th21;
end;
