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 image_Union: forall {A B: Type} (X Y: Ensemble A) (f: A -> B),
Same_set
(image_set (Union _ X Y) f)
(Union _ (image_set X f) (image_set Y f)).