proposition: Lemma plus_contin : forall (x y e : R), [0] [<] e -> {c : R | [0] [<] c | {d : R | [0] [<] d | forall x' y', AbsSmall c (x[-]x') -> AbsSmall d (y[-]y') -> AbsSmall e (x[+]y[-] (x'[+]y'))}}.
proof: Proof. intros. cut ([0] [<] e [/]TwoNZ). intro. exists (e [/]TwoNZ). auto. exists (e [/]TwoNZ). auto. intros. apply AbsSmall_wdr_unfolded with (x[-]x'[+](y[-]y')). apply AbsSmall_eps_div_two; auto. rational. apply div_resp_pos. apply pos_two. auto. Qed.
imports: CoRN.algebra.COrdAbs Coq(Lia)