The main aim of talk is twofold. Firstly, to present an elementary method based on Farkas' lemma for rationals on how to embed any finite partial subalgebra of a linearly ordered MV-algebra into $\mathbb Q\cap [0,1]$ and then to establish a new elementary proof of the completeness of the Lukasiewicz axioms. Secondly, to present a direct proof of Di Nola's Representation Theorem for MV-algebras and to extend his results to the restriction of the standard MV-algebra on rational numbers.