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_Empty: forall {A B: Type} (f: A -> B),
Same_set (image_set (Empty_set _) f) (Empty_set _).