  For k=1,2,... infty and a Frolicher-Kriegl order k Lipschitz differentiable
map f:E supseteq U to E having derivative at x_0 in U a linear homeomorphism E
to E and satisfying a Colombeau type tameness condition, we prove that x_0 has
a neighborhood V subseteq U with f|V a local order k Lipschitz diffeomorphism.
As a corollary we obtain a similar result for Keller C_c^{\infty} maps with E
in a class including Frechet and Silva spaces. We also indicate a procedure for
verifying the tameness condition for maps of the type x mapsto varphi circ
[id,x] and spaces E=C^{\infty}(Q) when Q is compact by considering the case
Q=[0,1]. Our considerations are motivated by the wish to try to retain
something valuable in an interesting but defective treatment of integrability
of Lie algebras by J. Leslie.

Functional Analysis