End Lindenbaum.
Definition Lindenbaum_preserves {A: Type} (P: Ensemble A -> Prop): Prop :=
forall (CA: Countable A) init, P init -> P (LindenbaumConstruction CA init P).
Definition Lindenbaum_ensures {A: Type} (P cP: Ensemble A -> Prop): Prop :=
forall (CA: Countable A) init, P init -> cP (LindenbaumConstruction CA init P).
Definition Lindenbaum_constructable {A: Type} (P cP: Ensemble A -> Prop): Prop :=
forall Phi, P Phi -> exists Psi: sig cP, Included _ Phi (proj1_sig Psi) /\ P (proj1_sig Psi).

Lemma Lindenbaum_preserves_by_finiteness {A: Type}: forall P: Ensemble A -> Prop,
finite_captured P ->
subset_preserved P ->
Lindenbaum_preserves P.