∀ {ι : Type u_1} {R : Type u_4} {M : Type u_2} {N : Type u_3} [inst : CommRing R] [inst_1 : AddCommGroup M]
  [inst_2 : Module R M] [inst_3 : AddCommGroup N] [inst_4 : Module R N] (P : RootPairing ι R M N) (f g : P.Aut),
  { val := ↑(f * g), inv := ↑(RootPairing.Equiv.symm P P (f * g)), val_inv := ⋯, inv_val := ⋯ } =
    { val := ↑f, inv := ↑(RootPairing.Equiv.symm P P f), val_inv := ⋯, inv_val := ⋯ } *
      { val := ↑g, inv := ↑(RootPairing.Equiv.symm P P g), val_inv := ⋯, inv_val := ⋯ }