unfold included in |- *.
simpl in |- *.
intros G A x H'; try assumption.
exists (Var (V:=A) (Build_subtype (E:=G) (P:=A) (subtype_elt:=x) H')); split;
[ idtac | try assumption ].
auto with algebra.
simpl in |- *.
auto with algebra.
Qed.

Lemma generated_group_prop :
forall (G : GROUP) (A : part_set G) (y : G),
in_part y (generated_group A) ->
exists x : FG A, Equal y (FG_lift (inj_part A) x).