The following statement is the first statement of the corollary in Section 27.4 of A. Kriegl, P.W. Michor, The convenient setting of global analysis:
If a (smoothly Hausdorff) smooth manifold is Lindelof and modeled on smoothly regular convenient vector spaces, then it is smoothly paracompact.
The proof only says "See (16.10)". By Theorem 16.10, which states that a Lindelof, smoothly regular space is smoothly paracompact, we have only to show that a smooth manifold as above is smoothly regular. But I cannot prove it.
How can we prove the following assertion?
If a (smoothly Hausdorff) smooth manifold is Lindelof and modeled on smoothly regular convenient vector spaces, then it is smoothly regular.