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_composition: forall {X Y Z:Type} (f:Y->Z) (g:X->Y)
(U:Ensemble Z), inverse_image (fun x:X => f (g x)) U =
inverse_image g (inverse_image f U).