Definition codeSysPrf :=
  codeSysPrf LNT codeLNTFunction codeLNTRelation codeArityLNTF codeArityLNTR
    codeArityLNTFIsPR codeArityLNTRIsPR (LNT2LNN_formula repT) v0.Definition codeSysPrfCorrect1 :=
  codeSysPrfCorrect1 LNT codeLNTFunction codeLNTRelation codeArityLNTF
    codeArityLNTR codeArityLNTFIsPR codeArityLNTFIsCorrect1 codeArityLNTRIsPR
    codeArityLNTRIsCorrect1 T' extendsNN T (LNT2LNN_formula repT) v0
    freeVarRepT' expressT'1.Definition codeSysPrfCorrect2 :=
  codeSysPrfCorrect2 LNT codeLNTFunction codeLNTRelation codeArityLNTF
    codeArityLNTR codeArityLNTFIsPR codeArityLNTFIsCorrect1 codeArityLNTRIsPR
    codeArityLNTRIsCorrect1 T' extendsNN T (LNT2LNN_formula repT) v0
    freeVarRepT' expressT'2.Definition codeSysPrfCorrect3 :=
  codeSysPrfCorrect3 LNT codeLNTFunction codeLNTRelation codeArityLNTF
    codeArityLNTR codeArityLNTFIsPR codeArityLNTFIsCorrect1
    codeArityLNTFIsCorrect2 codeArityLNTRIsPR codeArityLNTRIsCorrect1
    codeArityLNTRIsCorrect2 codeLNTFunctionInj codeLNTRelationInj T'
    extendsNN.Definition codePrf := codePrf LNT codeLNTFunction codeLNTRelation.Definition codeSysPrfNot :=
  codeSysPrfNot LNT codeLNTFunction codeLNTRelation codeArityLNTF
    codeArityLNTR codeArityLNTFIsPR codeArityLNTRIsPR 
    (LNT2LNN_formula repT) v0.Definition freeVarCodeSysPrfN :=
  freeVarCodeSysPrfN LNT codeLNTFunction codeLNTRelation codeArityLNTF
    codeArityLNTR codeArityLNTFIsPR codeArityLNTRIsPR 
    (LNT2LNN_formula repT) v0 freeVarRepT'.Definition codeSysPrfNCorrect1 :=
  codeSysPrfNCorrect1 LNT codeLNTFunction codeLNTRelation codeArityLNTF
    codeArityLNTR codeArityLNTFIsPR codeArityLNTFIsCorrect1 codeArityLNTRIsPR
    codeArityLNTRIsCorrect1 T' extendsNN T (LNT2LNN_formula repT) v0
    freeVarRepT' expressT'1.Definition codeSysPrfNCorrect2 :=
  codeSysPrfNCorrect2 LNT codeLNTFunction codeLNTRelation codeArityLNTF
    codeArityLNTR codeArityLNTFIsPR codeArityLNTFIsCorrect1 codeArityLNTRIsPR
    codeArityLNTRIsCorrect1 T' extendsNN T (LNT2LNN_formula repT) v0
    freeVarRepT' expressT'2.Definition codeSysPrfNCorrect3 :=
  codeSysPrfNCorrect3 LNT codeLNTFunction codeLNTRelation codeArityLNTF
    codeArityLNTR codeArityLNTFIsPR codeArityLNTFIsCorrect1
    codeArityLNTFIsCorrect2 codeArityLNTRIsPR codeArityLNTRIsCorrect1
    codeArityLNTRIsCorrect2 codeLNTFunctionInj codeLNTRelationInj T'
    extendsNN (LNT2LNN_formula repT) v0 freeVarRepT'.

Theorem PAIncomplete : exists f : Formula, (Sentence f) /\ ~(SysPrf PA f \/ SysPrf PA (notH f)).