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) holds (inferior_realsequence f)#x = inferior_realsequence R_EAL(f#x)
proof
  let f be Functional_Sequence of X,REAL;
  let x be Element of X;
  set F = inferior_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 Th4,MESFUNC5:def 13;
    hence (F#x).n = (inferior_realsequence R_EAL(f#x)).n by A1,Th4;
  end;
  hence thesis by FUNCT_2:63;
end;
