∀ {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)},
  H.prod K ≤ J ↔ AddSubgroup.map (AddMonoidHom.inl G N) H ≤ J ∧ AddSubgroup.map (AddMonoidHom.inr G N) K ≤ J