  The Zariski theorem says that for every hypersurface in a complex projective
(resp. affine) space of dimension at least 3 and for every generic plane in the
projective (resp. affine) space the natural embedding generates an isomorphism
of the fundamental groups of the complements to the hypersurface in the plane
and in the space. If a family of hypersurfaces depends algebraically on
parameters then it is not true in general that there exists a plane such that
the natural embedding generates isomorphisms of the fundamental groups of the
complements to each hypersurface from this family in the plane and in the
space. But we show that in the affine case such plane exists after a polynomial
coordinate substitution.
