  We prove two main results on Denjoy-Carleman classes: (1) a composite
function theorem which asserts that a function f(x) in a quasianalytic
Denjoy-Carleman class Q, which is formally composite with a generically
submersive mapping y=h(x) of class Q, at a single given point in the source (or
in the target) of h, can be written locally as f(x) = g(h(x)), where g(y)
belongs to a shifted Denjoy-Carleman class Q' ; (2) a statement on a similar
loss of regularity for functions definable in the o-minimal structure given by
expansion of the real field by restricted functions of quasianalytic class Q.
Both results depend on an estimate for the regularity of an infinitely
differentiable solution g of the equation f(x) = g(h(x)), with f and h as
above. The composite function result depends also on a quasianalytic
continuation theorem, which shows that the formal assumption at a given point
in (1) propagates to a formal composition condition at every point in a
neighbourhood.
