Definition wConsistent (T : System) :=
forall (f : Formula) (v : nat),
(forall x : nat, In x (freeVarFormula LNN f) -> v = x) ->
SysPrf T (existH v (notH f)) ->
exists n : nat, ~ SysPrf T (substituteFormula LNN f v (natToTerm n)).

(* Specifications: *)
Lemma wCon2Con : forall T : System, wConsistent T -> Consistent LNN T.
Definition wInconsistent (T : System) :=
exists f : Formula,
(exists v : nat,
(forall x : nat, In x (freeVarFormula LNN f) -> v = x) /\
SysPrf T (existH v (notH f)) /\
(forall n : nat, SysPrf T (substituteFormula LNN f v (natToTerm n)))).
Lemma notCon2wNotCon :
forall T : System, Inconsistent LNN T -> wInconsistent T.