Section Lindenbaum_Trivial.
Context {L: Language}
{minL: MinimumLanguage L}
{pL: PropositionalLanguage L}
{GammaP: Provable L}
{GammaD: Derivable L}
{SC: NormalSequentCalculus L GammaP GammaD}
{bSC: BasicSequentCalculus L GammaD}
{fwSC: FiniteWitnessedSequentCalculus L GammaD}
{minSC: MinimumSequentCalculus L GammaD}
{ipSC: IntuitionisticPropositionalSequentCalculus L GammaD}
{cpSC: ClassicalPropositionalSequentCalculus L GammaD}
{minAX: MinimumAxiomatization L GammaP}
{ipAX: IntuitionisticPropositionalLogic L GammaP}.

Lemma Lindenbaum_for_max_consistent: forall P,
Lindenbaum_ensures P derivable_closed ->
Lindenbaum_ensures P orp_witnessed ->
Lindenbaum_ensures P consistent ->
Lindenbaum_ensures P (maximal consistent).