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

theorem Th14:
  for f be Functional_Sequence of X,ExtREAL, 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,ExtREAL;
  let x be Element of X;
  assume that
A1: x in dom lim f and
A2: f#x is convergent;
A3: lim (f#x) = lim_inf (f#x) by A2,RINFSUP2:41;
A4: x in dom (f.0) by A1,Def9;
  then x in dom(lim_sup f) by Def8; then
A5: (lim_sup f).x = lim_sup (f#x) by Def8;
  x in dom(lim_inf f) by A4,Def7; then
A6: (lim_inf f).x = lim_inf (f#x) by Def7;
  lim (f#x) = lim_sup (f#x) by A2,RINFSUP2:41;
  hence thesis by A1,A5,A6,A3,Def9;
end;
