(* Because of later notation, this should also be known as: *)

Definition min_inj := group_inverse_inj.

(* Specifications: *)
Lemma subtype_elt_comp : forall (A : Setoid) (B : part_set A) (b b' : B), Equal b b' -> Equal (subtype_elt b) (subtype_elt b').
Lemma subtype_elt_inj : forall (A : Setoid) (B : part_set A) (b b' : B), Equal (subtype_elt b) (subtype_elt b') -> Equal b b'.
Lemma in_part_subtype_elt : forall (A : Setoid) (B : part_set A) (b : B), in_part (subtype_elt b) B.