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Dear Orbicular I think that even if you had a copy of Bourbaki on analitical and differentiable manifolds you couldn't find there what you search because it is just a collection od results.

For the theorem on the existence of slices , it 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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Dear Orbicular I think that even if you had a copy of Bourbaki on analitical and differentiable manifolds you couldn't find there what you search because it is just a collection od results.

For theorem on the existence of slices, it 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).