The problem of the sequential testing of hypotheses $H_i$, $i=1,\dots,N$, where hypotheses $H_i$ are described by the observed process (1.1), is considered. Let the assumption (1.3) be fulfilled and let $\pi_i(t)$ be defined by the formula (2.3), where $P_i(\,\cdot\,)$ is the measure corresponding to the observed process (1.1). It is proved that the decision rule (2.4) guarantees the asymptotically minimum expectation of time $\tau_\alpha^*$ of making the decision, if the probability of error $\alpha$ tends to zero. It is proved also that this optimal expectation of $\tau_\alpha^*$ satisfies the equation (2.6). It is found that the asymptotically optimal decision rule supplies the gain approximately in $4^\lambda$ times in comparison with the best nonsequential decision rule if $\alpha\ll 1$.