theorem Th18:
  for f be with_the_same_dom Functional_Sequence of X,REAL, E be
  Element of S st dom (f.0) = E & (for n be Nat holds f.n
  is E-measurable) holds lim_sup f is E-measurable
proof
  let f be with_the_same_dom Functional_Sequence of X,REAL, E be Element of S;
  assume that
A1: dom(f.0) = E and
A2: for n be Nat holds f.n is E-measurable;
  for n being Nat holds (R_EAL f).n is E-measurable 
  proof
    let n be Nat;
    f.n is E-measurable by A2;
    hence thesis by Th7;
  end;
  hence thesis by A1,MESFUNC8:23;
end;
