∀ {X : Type u_1} {M : Type u_2} [inst : TopologicalSpace X] [inst_1 : Monoid M] [inst_2 : TopologicalSpace M]
  [inst_3 : ContinuousMul M],
  Function.LeftInverse
    (fun f => { toFun := fun x => { val := ↑f x, inv := ↑f⁻¹ x, val_inv := ⋯, inv_val := ⋯ }, continuous_toFun := ⋯ })
    fun f =>
    { val := { toFun := fun x => ↑(f x), continuous_toFun := ⋯ },
      inv := { toFun := fun x => ↑(f x)⁻¹, continuous_toFun := ⋯ }, val_inv := ⋯, inv_val := ⋯ }