Claim: The Malgrange preparation theorem has two forms; one says that near the origina f can be writtend as the product of a smooth function c; the other Mather division is a sort of addition with remainder theorem.
Evidence: A second form of the theorem, occasionally called the Mather division theorem, is a sort of "division with remainder" theorem: it says that if f and k satisfy the conditions above and g is a smooth function near the origin, then we can write