  We solve fundamental problems in Oka theory by establishing an implicit
function theorem for sprays. As the first application of our implicit function
theorem, we obtain an elementary proof of the fact that approximation yields
interpolation. This proof and L\'{a}russon's elementary proof of the converse
give an elementary proof of the equivalence between approximation and
interpolation. The second application concerns the Oka property of a blowup. We
prove that the blowup of an algebraically Oka manifold along a smooth algebraic
center is Oka. In the appendix, equivariantly Oka manifolds are characterized
by the equivariant version of Gromov's condition $\mathrm{Ell}_{1}$, and the
equivariant localization principle is also given.
