Instance respectful_set_proper {A B: Type}: Proper (Same_set ==> pointwise_relation A (@eq B) ==> Same_set) respectful_set.
Instance image_set_proper2 {A B: Type}: Proper (Same_set ==> eq ==> Same_set) (@image_set A B).

Lemma resp_Intersection: forall {A B: Type} (X Y: Ensemble B) (f: A -> B),
Same_set
(respectful_set (Intersection _ X Y) f)
(Intersection _ (respectful_set X f) (respectful_set Y f)).