  We obtain a presentation by generators and relations of any Nichols algebra
of diagonal type with finite root system. We prove that the defining ideal is
finitely generated. The proof is based in Kharchenko's theory of PBW basis of
Lyndon words. We prove that the lexicographic order on Lyndon words is convex
for such PBW generators and so the PBW basis is orthogonal with respect to the
canonical non-degenerate form associated to the Nichols algebra.
