  We show that the Kuratowski imbedding of a Riemannian manifold in L^\infty,
exploited in Gromov's proof of the systolic inequality for essential manifolds,
admits an approximation by a (1+C)-bi-Lipschitz (onto its image),
finite-dimensional imbedding for every C>0. Our key tool is the first variation
formula thought of as a real statement in first-order logic, in the context of
non-standard analysis.
