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

theorem Th27:
  for f be Functional_Sequence of X,ExtREAL, g be PartFunc of X,ExtREAL st
  for x be Element of X st x in dom g holds
  f#x is convergent_to_finite_number &
    g.x = lim (f#x) holds g is real-valued
proof
  let f be Functional_Sequence of X,ExtREAL, g be PartFunc of X,ExtREAL;
  assume
A1: for x be Element of X st x in dom g holds
    f#x is convergent_to_finite_number & g.x = lim (f#x);
  now
    let x be Element of X;
    assume
A2: x in dom g; then
A3: not (lim (f#x)=+infty & f#x is convergent_to_+infty ) by A1,MESFUNC5:50;
    f#x is convergent_to_finite_number by A1,A2; then
A4: f#x is convergent by MESFUNC5:def 11;
    not (lim (f#x)=-infty & f#x is convergent_to_-infty ) by A1,A2,MESFUNC5:51;
    then consider g0 be Real such that
A5: lim (f#x) = g0 and
    for p be Real st 0<p ex n be Nat st for m be Nat st n<=m holds
    |.(f#x).m-lim (f#x).| < p and
    f#x is convergent_to_finite_number by A4,A3,MESFUNC5:def 12;
    |. g.x .| = |.g0 qua Complex.| by A1,A2,A5,EXTREAL1:12;
    hence |. g.x .| < +infty by XXREAL_0:9,XREAL_0:def 1;
  end;
  hence thesis by MESFUNC2:def 1;
end;
