reserve Omega for non empty set;
reserve r for Real;
reserve Sigma for SigmaField of Omega;
reserve P for Probability of Sigma;
reserve E for finite non empty set;

theorem Th6:
  for Omega be non empty finite set, f be PartFunc of Omega,REAL
  holds f is_simple_func_in Trivial-SigmaField (Omega) & dom f is Element of
  Trivial-SigmaField (Omega)
proof
  let Omega be non empty finite set, f be PartFunc of Omega,REAL;
  set Sigma = Trivial-SigmaField (Omega),D = dom f;
  ex F be Finite_Sep_Sequence of Sigma st dom f = union (rng F) & dom F =
dom (canFS(D)) &( for k be Nat st k in dom F holds F.k={(canFS(D) ).k})& for n
  being Nat for x,y being Element of Omega st n in dom F & x in F.n & y in F.n
  holds f.x = f.y by Lm6;
  hence thesis;
end;
