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;
