 So the existence of coordinate functions formula_16 is assured where formula_17 is an open covering for X.Then by the existence theorem there exists a Lie bundle formula_18 with the continuous mapping formula_19 inducing on each fibre the Lie bracket.Differential forms on "M" with values in formula_20 are in one-to-one correspondence with horizontal, "G"-equivariant Lie algebra-valued forms on "P".