(*********)
Definition adhDa (D:R -> Prop) (a:R) : Prop :=
  forall alp:R, alp > 0 ->  exists x : R, D x /\ Rdist x a < alp.

(* Specifications: *)
Lemma single_limit : forall (f:R -> R) (D:R -> Prop) (l l' x0:R), adhDa D x0 -> limit1_in f D l x0 -> limit1_in f D l' x0 -> l = l'.
Lemma limit_comp : forall (f g:R -> R) (Df Dg:R -> Prop) (l l' x0:R), limit1_in f Df l x0 -> limit1_in g Dg l' l -> limit1_in (fun x:R => g (f x)) (Dgf Df Dg f) l' x0.
Lemma limit_inv : forall (f:R -> R) (D:R -> Prop) (l x0:R), limit1_in f D l x0 -> l <> 0 -> limit1_in (fun x:R => / f x) D (/ l) x0.