Hint Resolve set_add_intro set_add_elim.

(* Specifications: *)
Lemma set_add_not_empty :
forall (a : A) (x : set), set_add a x <> empty_set.
Lemma set_union_intro1 :
forall (a : A) (x y : set), set_In a x -> set_In a (set_union x y).
Lemma set_union_intro2 :
forall (a : A) (x y : set), set_In a y -> set_In a (set_union x y).