Let X′ be a complex affine algebraic threefold with H3(X′) = 0 which is a UFD and whose invertible functions are constants. Let Z be 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 C2. We prove that X′ is isomorphic to C3 iff none of irreducible components of X′ \ Z has non-isolated singularities. Furthermore, if X′ is C3 then p extends to a polynomial on C3 which is linear in a suitable coordinate system. This implies the fact formulated in the title of the paper.
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