  We give a short proof that any smooth (means formally smooth and finitely
presented) homomorphism of rings can be obtained by base change from a smooth
homomorphism of noetherian rings. Together with the elegant short proof by J.
Conde-Lago that smooth homomorphisms of noetherian rings are flat, this gives a
short and elementary proof of the theorem of Grothendieck that all smooth
homomorphisms are flat.
