This paper is the final part of the syntactic demonstration of the Arithmetical Completeness of the modal system G; in the preceding parts [9] and [10] the tools for the proof were defined, in particular the notion of syntactic countermodel. Our strategy is: PA-completeness of G as a search for interpretations which force the distance between G and a GL-LIN-theorem to zero. If the GL-LIN-theorem S is not a G-theorem, we construct a formula H expressing the non G-provability of S, (...) so that ⊢GL-LIN ∼ H and so that a canonical proof T of ∼ H in GL-LIN is a syntactic countermodel for S with respect to G, which has the height θ(T) equal to the distance d(S, G) of S from G. Then we define the interpretation ξ of S which represents the proof-tree T in PA. By induction on θ(T), we prove that ⊢PA Sξ and d(S, G) > 0 imply the inconsistency of PA. (shrink)