So we can create a mapping from the partitioned

**I** onto U_i(X_i) by taking each index and associating it with the proper surjection (that is, with the aforementioned surjection which maps the indexes in a partition onto the union of {X_i} whose sets each have an index in that partition), which is unique since each index is in only one partition of

**I**. This is a surjection because, for any element x in one of the sets of {X_i}, it is in a set whose index is in a particular partition of

**I**, and since each partition of

**I** maps onto a subset of U_i(X_i), there is some i in

**I** that will map to that x under the surjection for its partition.