 given on each V_j a manifold structure such that for each pair j, j' the
| induced structure on V_j |^| V_j' coincides, then it is clear that we can
| give to X a unique manifold structure inducing the given ones on each V_j.
|
| If X, Y are two manifolds, then one can give the
| product X x Y a manifold structure in the obvious way.
| If {(U_i, q_i)} and {(V_j, r_j)} are atlases for X, Y
| respectively, then
|
| {(U_i x V_j, q_i x r_j)}
|
| is an atlas for the product, and the product of compatible
| atla