The well-definedness of the [connected sum][1] of two manifolds. After all, we choose two arbitrary balls for gluing; why should the result be independent? The proof depends on the nontrivial [Disc theorem][2].
 
I think that one usually uses this result as a black box.

  [1]: http://en.wikipedia.org/wiki/Connected_sum
  [2]: http://en.wikipedia.org/wiki/Disc_theorem