(** - ...and vice versa: if [x:(inject_subsets C)] then [(uninject_subsets x):C]

 %\label{uninjectsubsetsify}% *)

Definition uninject_subsetsify :
  forall (A : Setoid) (B : part_set A) (C : part_set B),
  inject_subsets C -> C.

Lemma in_part_included : forall (A : Setoid) (B C : part_set A) (a : A), in_part a B -> included B C -> in_part a C.