
theorem
  for X be non empty set, S be SigmaField of X, f be PartFunc of X,
  ExtREAL st f is_simple_func_in S
 ex F be Finite_Sep_Sequence of S, a be
  FinSequence of ExtREAL st F,a are_Re-presentation_of f
proof
  let X be non empty set;
  let S be SigmaField of X;
  let f be PartFunc of X,ExtREAL;
  assume f is_simple_func_in S;
  then consider
  F being Finite_Sep_Sequence of S, a be FinSequence of ExtREAL such
  that
A1: dom f = union rng F & dom F= dom a & for n be Nat st n in dom F for
  x be object st x in F.n holds f.x=a.n and
  for x be object st x in dom f
   ex ax be FinSequence of ExtREAL st dom ax= dom a &
  for n be Nat st n in dom ax
    holds ax.n=a.n*(chi(F.n,X)).x by Th3;
  take F,a;
  thus thesis by A1;
end;
