
theorem Th3:
  for X be non empty set, S be SigmaField of X, f be PartFunc of X,
  ExtREAL, M be sigma_Measure of S, F be Finite_Sep_Sequence of S, a,x be
  FinSequence of ExtREAL st f is_simple_func_in S & dom f <> {} &
  f is nonnegative &
F,a are_Re-presentation_of f & dom x = dom F &
(for n be Nat st n in dom x holds x.n=a.n*(M*F).n) holds integral(M,f)=Sum(
  x)
proof
  let X be non empty set;
  let S be SigmaField of X;
  let f be PartFunc of X,ExtREAL;
  let M be sigma_Measure of S;
  let F be Finite_Sep_Sequence of S;
  let a,x be FinSequence of ExtREAL;
A1: len F = len F;
  assume f is_simple_func_in S & dom f <> {} & f is nonnegative &
F,a are_Re-presentation_of f &( dom x = dom F & for n be Nat
  st n in dom x holds x.n=a.n*(M*F).n);
  hence thesis by A1,Th2;
end;
