Section Generated_module_def.
Variable R : RING.
Variable Mod : MODULE R.
Variable A : part_set Mod.
Definition generated_module : submodule Mod := coKer (FMd_lift (inj_part A)).
End Generated_module_def.
unfold included in |- *.
simpl in |- *.
intros R Mod A H H' x H'0; try assumption.
elim H'0; intros x0; clear H'0.
generalize x; clear x.
elim x0.
intros c; try assumption.
elim c.
simpl in |- *.
intros y subtype_prf x H'0; elim H'0; intros H'1 H'2; try exact H'2;
clear H'0.
apply in_part_comp_l with y; auto with algebra.
intros f H'0 f0 H'1 x H'2; elim H'2; intros H'3 H'4; try exact H'4; clear H'2.
simpl in H'4.
apply
in_part_comp_l
with
(sgroup_law Mod (FMd_lift_fun (inj_part A) f)
(FMd_lift_fun (inj_part A) f0)); auto with algebra.
simpl in |- *.
intros x H'0; elim H'0; intros H'1 H'2; try exact H'2; clear H'0.
apply in_part_comp_l with (monoid_unit Mod); auto with algebra.
intros f H'0 x H'1; try assumption.
elim H'1; intros H'2 H'3; simpl in H'3; clear H'1.
apply in_part_comp_l with (group_inverse Mod (FMd_lift_fun (inj_part A) f));
auto with algebra.
intros c f H'0 x H'1; try assumption.
simpl in H'1.
elim H'1; intros H'2 H'3; try exact H'3; clear H'1.
apply in_part_comp_l with (module_mult c (FMd_lift_fun (inj_part A) f));
auto with algebra.
Qed.
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.
simpl in |- *; auto with algebra.
intros R Mod A y H'; try assumption.
elim H'; intros x E; elim E; intros H'0 H'1; try exact H'1; clear E H'.
exists x; try assumption.
Qed.

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