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 Th3:
  for X being non empty set, f being Functional_Sequence of X,REAL
holds for x be Element of X st x in dom sup f holds (sup f).x = sup rng R_EAL(f
  #x)
proof
  let X be non empty set, f be Functional_Sequence of X,REAL;
  let x be Element of X;
  assume x in dom sup f;
  then (sup f).x = sup((R_EAL f)#x) by MESFUNC8:def 4;
  hence thesis by Th1;
end;
