
theorem Th14:
  for X be non empty set, S be SigmaField of X, f be PartFunc of X
,ExtREAL st f is_simple_func_in S & f is nonnegative
  holds ex F be Finite_Sep_Sequence of S, a be FinSequence of ExtREAL st F,a
  are_Re-presentation_of f & a.1=0. & for n be Nat st 2 <= n & n in dom a holds
  0. < a.n & a.n < +infty
proof
  let X be non empty set;
  let S be SigmaField of X;
  let f be PartFunc of X,ExtREAL;
  assume
A1: f is_simple_func_in S & f is nonnegative;
  per cases;
  suppose
    ex x be object st x in dom f & 0. = f.x;
    hence thesis by A1,Lm3;
  end;
  suppose
    for x be object st x in dom f holds 0. <> f.x;
    hence thesis by A1,Lm2;
  end;
end;
