proposition: Lemma add_zero_diff: forall x1 x2 y1 y2, x1 = x2 -> y1 ^ 2 = x1 ^ 3 + A * x1 + B -> y2 ^ 2 = x2 ^ 3 + A * x2 + B -> y1 <> -y2 -> y1 <>0.
proof: Proof. intros x1 x2 y1 y2 H H1 H2 H3 H4. assert (H5:= add_zero H H1 H2 H3). case H3; rewrite <- H5; ring [H4]. Qed.
imports: Arith_base Field_tac Ring Eqdep_dec FGroup List UList ZArith