proposition: Theorem ZOdiv_plus : forall a b c, (0 <= a * b)%Z -> (Z.quot (a + b) c = Z.quot a c + Z.quot b c + Z.quot (Z.rem a c + Z.rem b c) c)%Z.
proof: Proof. intros a b c Hab. destruct (Z.eq_dec c 0) as [Zc|Zc]. now rewrite Zc, 4!Zquot_0_r. apply Zmult_reg_r with (1 := Zc). rewrite 2!Zmult_plus_distr_l. assert (forall d, Z.quot d c * c = d - Z.rem d c)%Z. intros d. rewrite ZOmod_eq. ring. rewrite 4!H. rewrite <- Zplus_rem with (1 := Hab). ring. Qed.
imports: Coq(ZArith Lia Zquot)