We prove here that in the Theorem on Local Ergodicity for Semi-Dispersive
Billiards (proved by N. I. Chernov and Ya. G. Sinai in 1987) the condition of
the so called ``Ansatz'' can be dropped. That condition assumed that almost
every singular phase point had a hyperbolic trajectory after the singularity.
Having this condition dropped, the cited theorem becomes much stronger and
easier to apply. At the end of the paper two immediate corollaries of this
improvement are discussed: One of them is the (fully hyperbolic) Bernoulli
mixing property of every hard disk system (D=2), the other one claims that the
ergodic components of every hard ball system ($D\ge3$) are open.