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

theorem
  for f be Functional_Sequence of X,ExtREAL, x be Element of X st
  x in dom (f.0) holds f#x is convergent iff (lim_sup f).x = (lim_inf f).x
proof
  let f be Functional_Sequence of X,ExtREAL, x be Element of X;
  assume
A1: x in dom(f.0);
  then x in dom(lim_inf f) by Def7;
  then
A2: (lim_inf f).x = lim_inf (f#x) by Def7;
  x in dom(lim_sup f) by A1,Def8;
  then (lim_sup f).x = lim_sup (f#x) by Def8;
  hence thesis by A2,RINFSUP2:40;
end;
