∀ {M : Type u_5} [inst : CommMonoid M] [inst_1 : PartialOrder M] [WellQuasiOrderedLE M] [IsOrderedCancelMonoid M]
  [CanonicallyOrderedMul M] {P : Submonoid M}, (∀ x ∈ P, ∀ (y : M), x * y ∈ P → y ∈ P) → P.FG