Definition FiniteT_nat_cardinal (X:Type) (H:FiniteT X) : nat :=
  proj1_sig (constructive_definite_description _
              (FiniteT_has_nat_cardinal X H)).

(* Specifications: *)
Lemma FiniteT_nat_cardinal_def: forall (X:Type) (H:FiniteT X), cardinal _ (@Full_set X) (FiniteT_nat_cardinal X H).
Lemma FiniteT_nat_cardinal_cond: forall (X:Type) (H:FiniteT X) (n:nat), cardinal _ (@Full_set X) n -> FiniteT_nat_cardinal X H = n.
Lemma FiniteT_nat_cardinal_False: FiniteT_nat_cardinal False empty_finite = 0.
Lemma injection_preserves_cardinal: forall (X Y:Type) (f:X->Y) (n:nat) (S:Ensemble X), cardinal _ S n -> injective f -> cardinal _ (Im S f) n.
Lemma FiniteT_nat_cardinal_option: forall (X:Type) (H:FiniteT X), FiniteT_nat_cardinal (option X) (add_finite X H) = S (FiniteT_nat_cardinal X H).
Lemma FiniteT_nat_cardinal_bijection: forall (X Y:Type) (H:FiniteT X) (g:X->Y) (Hinv:invertible g), FiniteT_nat_cardinal Y (bij_finite X Y g H Hinv) = FiniteT_nat_cardinal X H.
Lemma unique_FiniteT_nat_cardinal: exists! f: (forall (X:Type), FiniteT X -> nat), f False empty_finite = 0 /\ (forall (X:Type) (H:FiniteT X), f (option X) (add_finite X H) = S (f X H)) /\ (forall (X Y:Type) (H:FiniteT X) (g:X->Y) (Hinv:invertible g), f Y (bij_finite X Y g H Hinv) = f X H).