proposition: Lemma regFunEq_e : forall (f g : RegularFunction (@ball X)), (forall e : Qpos, ball (m:=X) (proj1_sig e + proj1_sig e) (approximate f e) (approximate g e)) -> (regFunEq f g).
proof: Proof. unfold regFunEq. intros f g H e1 e2. apply ball_closed. intros d dpos. setoid_replace (proj1_sig e1 + proj1_sig e2 + d) with ((proj1_sig e1 + ((1#4) *d) + (((1#4)*d) + ((1#4)*d)) +(((1#4)*d)+proj1_sig e2))) by (simpl; ring). eapply ball_triangle. eapply ball_triangle. apply (regFun_prf _ e1 ((1#4)*exist _ _ dpos)%Qpos). apply (H ((1 # 4) * exist (Qlt 0) d dpos)%Qpos). apply (regFun_prf _ ((1 # 4) * exist (Qlt 0) d dpos)%Qpos e2). Qed.
imports: CoRN.metric2.UniformContinuity CoRN.model.structures.QposInf CoRN.metric2.Classification CoRN.model.totalorder.QposMinMax CoRN.model.totalorder.QMinMax