  We consider a one-loop finite ${\cal N}=1$ supersymmetric theory in such a
renormalization scheme that the first $L$ contributions to the gauge
$\beta$-function and the first $(L-1)$ contributions to the anomalous dimension
of the matter superfields and to the Yukawa $\beta$-function vanish. It is
demonstrated that in this case the NSVZ equation and the exact equation for the
Yukawa $\beta$-function in the first nontrivial order are valid for an
arbitrary renormalization prescription respecting the above assumption. This
implies that under this assumption the $(L+1)$-loop contribution to the gauge
$\beta$-function and the $L$-loop contribution to the Yukawa $\beta$-function
are always expressed in terms of the $L$-loop contribution to the anomalous
dimension of the matter superfields. This statement generalizes the result of
Grisaru, Milewski, and Zanon that for a theory finite in $L$ loops the
$(L+1)$-loop contribution to the $\beta$-function also vanishes. In particular,
it gives a simple explanation why their result is valid although the NSVZ
equation does not hold in an arbitrary subtraction scheme.
