Require Export FunctionProperties.
Require Export Ensembles.
Require Export EnsemblesSpec.
Definition inverse_image {X Y:Type} (f:X->Y) (T:Ensemble Y) : Ensemble X :=
[ x:X | In T (f x) ].
Hint Unfold inverse_image : sets.

Lemma inverse_image_union: forall {X Y:Type} (f:X->Y)
(T1 T2:Ensemble Y), inverse_image f (Union T1 T2) =
Union (inverse_image f T1) (inverse_image f T2).