Definition is_ideal (I : subgroup R) :=
  forall x : R, in_part x I -> forall a : R, in_part (ring_mult a x) I.Record ideal : Type := 
  {ideal_subgroup :> subgroup R; ideal_prf : is_ideal ideal_subgroup}.Record ideal : Type := 
  {ideal_subgroup :> subgroup R; ideal_prf : is_ideal ideal_subgroup}.Record ideal : Type := 
  {ideal_subgroup :> subgroup R; ideal_prf : is_ideal ideal_subgroup}.

Lemma ideal_prop4 : forall (I : ideal) (x : R), in_part x I -> in_part (group_inverse R x) I.