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.

Lemma generated_group_minimal :
forall (G : GROUP) (A : part_set G) (H : subgroup G),
included A H -> included (generated_group A) H.