Instance Included_proper (V: Type): Proper (Same_set ==> Same_set ==> iff) (@Included V).Instance complement_proper (V: Type): Proper (Same_set ==> Same_set) (Complement V).Instance Union_proper (V: Type): Proper (Same_set ==> Same_set ==> Same_set) (Union V).Instance Disjoint_proper (V: Type): Proper (Same_set ==> Same_set ==> iff) (Disjoint V).

Lemma Union_Included {A: Type}: forall P Q R, Included (Union A P Q) R <-> Included P R /\ Included Q R.