 in Y there exists a chart (V, r) at y
| such that r gives an isomorphism of V with a product V_1 x V_2 where
| V_1 is open in some space E_1 and V_2 is open in some space E_2,
| and such that
|
| r(Y |^| V) = V_1 x a_2
|
| for some point a_2 in V_2 (which we could take to be 0).  Then it is clear
| that Y is locally closed in X.  Furthermore, the map r induces a bijection
|
| r_1 : Y |^| V -&gt; V_1.
|
| The collection of pairs (Y |^| V, r_1) obtained in the above manner constitues
| an atlas for Y, of cla