  The classical lemma of Borel reads: any power series with real coefficients
is the Taylor series of a smooth function. Algebraically this means the
surjectivity of the completion map at a point, $C^\infty(\Bbb{R}^n)
\twoheadrightarrow \Bbb{R}[[\underline{x}]]$. Similarly, Whitney extension
theorem implies the surjectivity of the completion at closed subsets of
$\Bbb{R}^n$.
  For various applications one needs the surjectivity of completion for general
$C^\infty$-rings and general filtrations. We establish the necessary and
sufficient conditions for this surjectivity.
  Moreover, we prove: any element of the completion admits a
$C^\infty$-representative that is real-analytic outside of the locus of
completion, has any prescribed vanishing rate "at infinity", and the prescribed
positivity behaviour at the finite part. Alternatively, one can impose on the
smooth representative a set of (compatible) linear conditions.
