Borel's lemma reads: any power series with real coefficients is realizable as the
Taylor series of a smooth function.
Algebraically this means the surjectivity of the completion map,
C^\infty> R[[x]], for the completion with respect to the powers of the maximal
ideal.
For various applications (e.g. in Singularities) one needs the surjectivity of
completion for more general filtrations, not necessarily of the form {I^j}.
We prove the necessary/sufficient conditions for this surjectivity.
This can be regarded as an algebraic version of the classical Whitney's extension
theorem.
