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 Th11:
  for f be Functional_Sequence of X,REAL, x be Element of X st x
in dom(f.0) holds (superior_realsequence f)#x = superior_realsequence R_EAL(f#x
  )
proof
  let f be Functional_Sequence of X,REAL, x be Element of X;
  set F = superior_realsequence f;
  assume
A1: x in dom(f.0);
  now
    let n be Element of NAT;
    dom(F.n) = dom(f.0) & (F#x).n = (F.n).x by Th5,MESFUNC5:def 13;
    hence (F#x).n =(superior_realsequence R_EAL(f#x)).n by A1,Th5;
  end;
  hence thesis by FUNCT_2:63;
end;
