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 Th15:
  for f be Functional_Sequence of X,REAL, x be Element of X st x
in dom lim f & f#x is convergent holds (lim f).x= (lim_sup f).x & (lim f).x = (
  lim_inf f).x
proof
  let f be Functional_Sequence of X,REAL;
  let x be Element of X;
  assume that
A1: x in dom lim f and
A2: f#x is convergent;
  R_EAL(f#x) is convergent by A2,RINFSUP2:14;
  then
A3: lim R_EAL(f#x) = lim_sup R_EAL(f#x) & lim R_EAL(f#x) = lim_inf R_EAL(f#x
  ) by RINFSUP2:41;
A4: x in dom (f.0) by A1,MESFUNC8:def 9;
  then x in dom lim_inf f by MESFUNC8:def 7;
  then
A5: (lim_inf f).x = lim_inf R_EAL(f#x) by Th12;
  x in dom lim_sup f by A4,MESFUNC8:def 8;
  then (lim_sup f).x = lim_sup R_EAL(f#x) by Th13;
  hence thesis by A1,A5,A3,Th14;
end;
