Technically, an elliptic curve can only be constructed over a field, and formula_4 is only a field if "n" is prime, so we seem to be assuming the result we're trying to prove. The difficulty arises in the elliptic-curve addition algorithm, which takes inverses in the field that may not exist in formula_4. However, it can be shown (lemma 1 of "Almost All Primes Can Be Quickly Certified") that if we merely perform computations as though the curve were well-defined and do not at any point attempt to invert an element with no inverse, the result is still valid; if we do encounter an element with no inverse, this establishes that "n" is composite.