∀ {X : Type u_1} [inst : TopologicalSpace X] {x₀ x₁ : X} (p : Path x₀ x₁) (f : ↑unitInterval → ↑unitInterval)
  (hf : Continuous f),
  f 0 = 0 →
    f 1 = 1 →
      ∀ (t x : ↑unitInterval),
        x ∈ {0, 1} →
          { toFun := fun x => p ⟨↑(unitInterval.symm (t, x).1) * ↑(t, x).2 + ↑(t, x).1 * ↑(f (t, x).2), ⋯⟩,
                continuous_toFun := ⋯ }
              x =
            p.toContinuousMap x