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
  for Omega be non empty finite set, f be PartFunc of Omega,REAL holds
ex F be Finite_Sep_Sequence of Trivial-SigmaField (Omega), s being FinSequence
of (dom f) st dom f = union (rng F) & dom F = dom (s) & s is one-to-one & rng s
= dom f & len s = card (dom f) & (for k be Nat st k in dom F holds F.k={s.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
proof
  let Omega be non empty finite set, f be PartFunc of Omega,REAL;
  set Sigma = Trivial-SigmaField (Omega);
  set D = dom f;
  set s=canFS(D);
A1: len s = card (dom f) by FINSEQ_1:93;
  (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 )& rng s = dom f by Lm6,FUNCT_2:def 3;
  hence thesis by A1;
end;
