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Definition is_gcd (a b d : S) :=
  divide d a /\
  divide d b /\ (forall q : S, divide q a -> divide q b -> divide q d).

Lemma simplification_integrity : is_unitary_commutative_ring -> integrity -> forall a x : S, A a -> A x -> a <> O -> Mult a x = a -> x = I.