  We prove an analogue of the de Rham theorem for the extended L^2-cohomology
introduced by M. Farber. This is done by establishing that the de Rham complex
over a compact closed manifold with coefficients in a flat Hilbert bundle E of
A-modules over a finite von Neumann algebra A is chain-homotopy equivalent
(with bounded morphisms and homotopy operators) to a combinatorial complex with
the same coefficients. This is established by using the Witten deformation of
the de Rham complex. We also prove that the de Rham complex is chain-homotopy
equivalent to the spectrally truncated de Rham complex which is also finitely
generated.
