  Suppose that X' is a smooth affine algebraic variety of dimension 3 with
H_3(X')=0 which is a UFD and whose invertible functions are constants. Suppose
that Z is a Zariski open subset of X which has a morphism p : Z -> U into a
curve U such that all fibers of p are isomorphic to C^2. We prove that X' is
isomorphic to C^3 iff none of irreducible components of X'-Z has non-isolated
singularities. Furthermore, if X' is C^3 then p extends to a polynomial on C^3
which is linear in a suitable coordinate system. As a consequence we obtain the
fact formulated in the title of the paper.
