(* the number of different elements in X is finite *)
  Inductive cardinality {X} (P:X->Prop): nat -> Prop:=
  |Cardinality n (f:nat-> X): (forall i j, i<n -> j<n -> i<>j -> f(i)<>f(j)) ->
                           (forall i, i<n -> P (f i)) ->
                           (forall x, P x -> exists i, i < n /\ f i = x) ->
                           cardinality P n.(* the number of different elements in X is finite *)
  Inductive cardinality {X} (P:X->Prop): nat -> Prop:=
  |Cardinality n (f:nat-> X): (forall i j, i<n -> j<n -> i<>j -> f(i)<>f(j)) ->
                           (forall i, i<n -> P (f i)) ->
                           (forall x, P x -> exists i, i < n /\ f i = x) ->
                           cardinality P n.Inductive cardinality' {X} (P:X->Prop): nat -> Prop:=
  |Cardinality' n (f:nat-> X): (forall i j, i<n -> j<n -> i<>j  -> f(i)<>f(j)) ->
                            (forall i, i<n -> P (f i)) ->
                            cardinality' P n.Inductive cardinality' {X} (P:X->Prop): nat -> Prop:=
  |Cardinality' n (f:nat-> X): (forall i j, i<n -> j<n -> i<>j  -> f(i)<>f(j)) ->
                            (forall i, i<n -> P (f i)) ->
                            cardinality' P n.

Lemma finite_cardinality: forall (P:X->Prop) (EM: forall x, P x \/ ~ P x), finite P -> exists n, cardinality P n.