This is a report on very recent work in progress (joint with H-Y Liao and P Xu) proving the theorem in the title. The purpose is to embed differentiable manifolds in a context where homotopy theory is possible: for example, the fibre products and intersections of differentiable manifolds, which do not exist in the category of differentiable manifolds, do exist as homotopy fibered products in the category of fibrant objects I will describe. Furthermore, deformation theory, by which I mean the homotopy theory of differential graded Lie algebras, is embedded in this context as well. The hope is that this will lead to a simpler context for derived differentiable topology than other more involved constructions. The two talks will be aimed at a general audience. The proof will be an application of the transfer theorem for curved L-infinity algebras, to which these lectures can serve as an introduction.