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

(* Specifications: *)
Lemma generated_module_prop :
forall (R : RING) (Mod : MODULE R) (A : part_set Mod) (y : Mod),
in_part y (generated_module A) ->
exists x : FMd R A, Equal y (FMd_lift (inj_part A) x).