  We derive the existence of Hopf subalgebras generated by Green's functions in
the Hopf algebra of Feynman graphs of a quantum field theory. This means that
the coproduct closes on these Green's functions. It allows us for example to
derive Dyson's formulas in quantum electrodynamics relating the renormalized
and bare proper functions via the renormalization constants and the analogous
formulas for non-abelian gauge theories. In the latter case, we observe the
crucial role played by Slavnov--Taylor identities.
