Section Generated_group_def.
Variable G : GROUP.
Variable A : part_set G.
Definition generated_group : subgroup G := coKer (FG_lift (inj_part A)).
End Generated_group_def.
unfold included in |- *.
simpl in |- *.
intros G 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 G (FG_lift_fun (inj_part A) f) (FG_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 G); 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 G (FG_lift_fun (inj_part A) f));
auto with algebra.
Qed.
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).