Definition RepresentsInSelf (T:System) := 
exists rep:Formula, exists v:nat,
(forall x : nat, In x (freeVarFormula LNN rep) -> x = v)  /\
(forall f : Formula,
        mem Formula T f ->
        SysPrf T (substituteFormula LNN rep v (natToTerm (codeFormula f)))) /\
(forall f : Formula,
        ~ mem Formula T f ->
        SysPrf T
          (notH (substituteFormula LNN rep v (natToTerm (codeFormula f))))).Definition DecidableSet (A:_)(s:Ensemble A) :=
(forall x : A,
        mem A s x \/
        ~ mem A s x).

(* Specifications: *)
Theorem Incompleteness : forall T : System, Included Formula NN T -> RepresentsInSelf T -> DecidableSet Formula T -> exists f : Formula, (Sentence f) /\ (SysPrf T f \/ SysPrf T (notH f) -> Inconsistent LNN T).