End Domin.
(** is_equiv is compatible with the vector space structure *)
Section Equiv_VS.
Context {T : Type} {K : AbsRing} {V : NormedModule K}.

(* Specifications: *)
Lemma equiv_scal :
forall {F : (T -> Prop) -> Prop} {FF : Filter F} (f g : T -> V) (c : K),
(exists y : K, mult y c = one) ->
is_equiv F f g -> is_equiv F (fun x => scal c (f x)) (fun x => scal c (g x)).
Lemma equiv_plus :
forall {F : (T -> Prop) -> Prop} {FF : Filter F} (f o : T -> V),
is_domin F f o -> is_equiv F (fun x => plus (f x) (o x)) f.