and reduce the question to one of maps
|         of open subsets of Banach spaces.  In that case, the conclusion concerning the
|         tangent spaces follows at once from the assumed direct product decompositions.
|         Conversely, assume our condition on the tangent map.  The question being local,
|         we can assume that Y = V_1 x V_2 is a product of open sets in Banach spaces
|         such that W = V_1 x 0, and we can also assume that X = U is open in some
|         Banach space, x = 0.  Then