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I am looking for a reference treating the slice theorem for Banach Lie group actions on Banach manifolds, i.e. proving that a smooth, free and proper action of a Banach Lie group $G$ on a Banach manifold $M$ with embedded orbits ensures that the quotient $M/G$ inherits the structure of a Banach manifold, s.t. $M\rightarrow M/G$ becomes a $G$-principal bundle.

I know this is briefly treated in Bourbaki's "Lie groups and Lie algebras". Unfortunately, in the proof they refer to the Bourbaki book "differentiable and analytic manifolds" which is not available to me.

Could anyone please provide me with a reference?

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Dear Orbicular the theorem on the existence of slices is stated without proof as Theorem 5.2.6 in Critical Point Theory and Submanifold Geometry, LNM 1353, of Palais and Terng (for example see here).
The proof should be adapted without difficulty from that in the finite-dimensional case.
For this case you can look at ``On the existence of slices for actions of non-compact groups'' by Palais (for example see here).

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