  Confirming previous heuristic analyses \`a la Belinskii-Khalatnikov-Lifshitz,
it is rigorously proven that certain ``subcritical'' Einstein-matter systems
exhibit a monotone, generalized Kasner behaviour in the vicinity of a spacelike
singularity. The D-dimensional coupled Einstein-dilaton-p-form system is
subcritical if the dilaton couplings of the p-forms belong to some dimension
dependent open neighbourhood of zero, while pure gravity is subcritical if D is
greater than or equal to 11. Our proof relies, like the recent theorem dealing
with the (always subcritical) Einstein-dilaton system, on the use of Fuchsian
techniques, which enable one to construct local, analytic solutions to the full
set of equations of motion. The solutions constructed are ``general'' in the
sense that they depend on the maximal expected number of free functions.
