  The celebrated Morlet-Burghelea-Lashof-Kirby-Siebenmann smoothing theory
theorem states that the group $\mathrm{Diff}_\partial(D^n)$ of diffeomorphisms
of a disc $D^n$ relative to the boundary is equivalent to
$\Omega^{n+1}\left(\mathrm{PL}_n/\mathrm{O}_n\right)$ for any $n\geq 1$ and to
$\Omega^{n+1}\left(\mathrm{TOP}_n/\mathrm{O}_n\right)$ for $n\neq 4$. We revise
smoothing theory results to show that the delooping generalizes to different
versions of disc smooth embedding spaces relative to the boundary, namely the
usual embeddings, those modulo immersions, and framed embeddings. The latter
spaces deloop as
$\mathrm{Emb}_\partial^{fr}(D^m,D^n)\simeq\Omega^{m+1}\left(\mathrm{O}_n\backslash\!\!\backslash\mathrm{PL}_n/\mathrm{PL}_{n,m}\right)\simeq
\Omega^{m+1}\left(\mathrm{O}_n\backslash\!\!\backslash\mathrm{TOP}_n/\mathrm{TOP}_{n,m}\right)$
for any $n\geq m\geq 1$ ($n\neq 4$ for the second equivalence), where the
left-hand side in the case $n-m=2$ or $(n,m)=(4,3)$ should be replaced by the
union of the path-components of $\mathrm{PL}$-trivial knots (framing being
disregarded). Moreover, we show that for $n\neq 4$, the delooping is compatible
with the Budney $E_{m+1}$-action. We use this delooping to combine the Hatcher
$\mathrm{O}_{m+1}$-action and the Budney $E_{m+1}$-action into a framed little
discs operad $E_{m+1}^{\mathrm{O}_{m+1}}$-action on
$\mathrm{Emb}_\partial^{fr}(D^m,D^n)$.
