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 Th14:
  for X be non empty set, f be Functional_Sequence of X,REAL holds
  for x be Element of X st x in dom lim f holds (lim f).x=lim R_EAL(f#x)
proof
  let X be non empty set, f be Functional_Sequence of X,REAL;
  let x be Element of X;
  assume x in dom lim f;
  then (lim f).x = lim((R_EAL f)#x) by MESFUNC8:def 9;
  hence thesis by Th1;
end;
