∀ {G : Type u_1} [inst : AddGroup G] {N : Type u_5} [inst_1 : AddGroup N] {H : AddSubgroup G} {K : AddSubgroup N}
  {J : AddSubgroup (G × N)},
  J ≤ H.prod K ↔ AddSubgroup.map (AddMonoidHom.fst G N) J ≤ H ∧ AddSubgroup.map (AddMonoidHom.snd G N) J ≤ K