(** - Once we have [inject_subsets], we may need to view an element of [C] as an element
of the injected subset: so, if [x:C] then [(inject_subsetsify x):(inject_subsets C)]

 %\label{injectsubsetsify}% *)

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

Lemma inject_subsetsify_comp : forall (A : Setoid) (B : part_set A) (C : part_set B) (c c' : C), Equal c c' -> Equal (inject_subsetsify c) (inject_subsetsify c').