Since the formal deductive system L{}*was built up in 1997, it has played important roles in the theoretical and applied research of fuzzy logic and fuzzy reasoning. But, up to now, the completeness problem of the system L{}* is still an open problem. In this paper, the properties and structure of R0 algebras are further studied, and it is shown that every tautology on the R0 interval [0,1] is also a tautology on any R0 algebra. Furthermore, based on the particular structure of L{}*_Lindenbaum algebra, the completeness and strong completeness of the system L{}* are proved. Some applications of the system L{}* in fuzzy reasoning are also discussed, and the obtained results and examples show that the system L{}* is suprior to some other important fuzzy logic systems.