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
  for f be Functional_Sequence of X,REAL, x be Element of X st x in dom
  (f.0) & f#x is convergent holds (superior_realsequence f)#x is bounded_below
proof
  let f be Functional_Sequence of X,REAL, x be Element of X;
  assume
A1: x in dom (f.0);
  assume f#x is convergent;
  then
A2: f#x is bounded;
  then superior_realsequence R_EAL(f#x) = superior_realsequence (f#x) by
RINFSUP2:9;
  then
A3: (superior_realsequence f)#x = superior_realsequence(f#x) by A1,Th11;
  superior_realsequence(f#x) is bounded by A2,RINFSUP1:56;
  hence thesis by A3,RINFSUP2:13;
end;
