proposition: Lemma limit_div_pos_ext : forall f g (I : R -> Prop) a, (forall x, I x -> f x = g x) -> limit_div_pos f I a -> limit_div_pos g I a.
proof: Proof. intros. unfold limit_div_neg in *. intros m Hm. specialize (H0 m Hm). destruct H0 as [alp [Halp Hsolve]]. exists alp. split. assumption. intros. specialize (Hsolve x H0 H1). rewrite <- H. apply Hsolve. apply H0. Qed.
imports: Reals Lra Rfunction_classes_def Cauchy_lipschitz Rextensionality