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In this note we show that for any proper action of a Banach--Lie group $G$ ona Banach manifold $M$, the corresponding tangent maps $\g \to T_x(M)$ haveclosed range for each $x \in M$, i.e., the tangent spaces of the orbits areclosed. As a consequence, for each free proper action on a Hilbert manifold,the quotient $M/G$ carries a natural manifold structure.

Closedness of the tangent spaces to the orbits of proper actions