  In "A new proof of the completeness of the Lukasiewicz axioms"} (Transactions
of the American Mathematical Society, 88) C.C. Chang proved that any totally
ordered $MV$-algebra $A$ was isomorphic to the segment $A \cong \Gamma(A^*, u)$
of a totally ordered $l$-group with strong unit $A^*$. This was done by the
simple intuitive idea of putting denumerable copies of $A$ on top of each other
(indexed by the integers). Moreover, he also show that any such group $G$ can
be recovered from its segment since $G \cong \Gamma(G, u)^*$, establishing an
equivalence of categories. In "Interpretation of AF $C^*$-algebras in
Lukasiewicz sentential calculus" (J. Funct. Anal. Vol. 65) D. Mundici extended
this result to arbitrary $MV$-algebras and $l$-groups with strong unit. He
takes the representation of $A$ as a sub-direct product of chains $A_i$, and
observes that $A \overset {} {\hookrightarrow} \prod_i G_i$ where $G_i =
A_i^*$. Then he let $A^*$ be the $l$-subgroup generated by $A$ inside $\prod_i
G_i$. He proves that this idea works, and establish an equivalence of
categories in a rather elaborate way by means of his concept of good sequences
and its complicated arithmetics. In this note, essentially self-contained
except for Chang's result, we give a simple proof of this equivalence taking
advantage directly of the arithmetics of the the product $l$-group $\prod_i
G_i$, avoiding entirely the notion of good sequence.
