From Coq Require Export Ensembles.
From ZornsLemma Require Export Image.
From Coq Require Export Finite_sets.
From ZornsLemma Require Export FunctionProperties.
Set Asymmetric Patterns.
Inductive FiniteT : Type -> Prop :=
| empty_finite: FiniteT False
| add_finite: forall T:Type, FiniteT T -> FiniteT (option T)
| bij_finite: forall (X Y:Type) (f:X->Y), FiniteT X ->
invertible f -> FiniteT Y.

Lemma finite_dep_choice: forall (A:Type) (B:forall x:A, Type)
(R:forall x:A, B x->Prop),
FiniteT A -> (forall x:A, exists y:B x, R x y) ->
exists f:(forall x:A, B x), forall x:A, R x (f x).