  For a geodesic ball with non-negative Ricci curvature and almost maximal
volume, without using compactness argument, we construct an
$\epsilon$-splitting map on a concentric geodesic ball with uniformly small
radius. There are two new technical points in our proof. The first one is the
way of finding $n$ directional points by induction and stratified almost Gou-Gu
Theorem. The other one is the error estimates of projections, which guarantee
the $n$ directional points we find really determine $n$ different directions.
