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 Th5:
  for X be non empty set, f being Functional_Sequence of X,REAL, n
being Nat holds dom((superior_realsequence f).n) = dom(f.0) & for x
  be Element of X st x in dom((superior_realsequence f).n) holds ((
  superior_realsequence f).n).x=(superior_realsequence R_EAL(f#x)).n
proof
  let X be non empty set;
  let f be Functional_Sequence of X,REAL;
  let n be Nat;
  set SF = superior_realsequence f;
  thus dom((superior_realsequence f).n) = dom(f.0) by MESFUNC8:def 6;
  hereby
    let x be Element of X;
    assume x in dom(SF.n);
    then (SF.n).x = (superior_realsequence((R_EAL f)#x)).n by MESFUNC8:def 6;
    hence (SF.n).x = (superior_realsequence R_EAL(f#x)).n by Th1;
  end;
end;
