Is Total Time Derivative Velocity

lemma isTotalTimeDerivativeVelocity  [CompleteSpace X]    (δL : X → ℝ)    (hδL0 : δL 0 = 0)    (h : IsTotalTimeDerivative (fun _ _ v => δL v)) :    ∃ g : X, ∀ v, δL v = ⟪g, v⟫_ℝ := by  classical  rcases (isTotalTimeDerivative_explicit.mp h) with ⟨F, hFdiff, hEq⟩   -- Derivative of F at (0,0)  let dF : (Time  × X) →L[ℝ] ℝ :=    fderiv ℝ ↿F ((0 : Time), (0 : X))   -- The "time-direction" derivative must vanish because δL 0 = 0.  have h_time : dF ((1 : Time), (0 : X)) = 0 := by    have h0 :        δL (0 : X) =          fderiv ℝ ↿F ((0 : Time), (0 : X))            ((1 : Time), (0 : X)) := by      simpa using (hEq (0 : Time) (0 : X)        (0 : X))    have : dF ((1 : Time), (0 : X)) =        δL (0 : X) := by      simpa [dF] using h0.symm    simpa [hδL0] using this   -- Induced continuous linear functional on velocity: v ↦ dF (0,v).  let φ : X →L[ℝ] ℝ :=    dF.comp (ContinuousLinearMap.inr ℝ Time X)   -- Show δL v = φ v for all v.  have hφ : ∀ v : X, δL v = φ v := by    intro v    have hv :        δL v =          fderiv ℝ ↿F ((0 : Time), (0 : X))            ((1 : Time), v) := by      simpa using (hEq (0 : Time) (0 : X) v)    have hv' : δL v = dF ((1 : Time), v) := by      simpa [dF] using hv    calc      δL v = dF ((1 : Time), v) := hv'     _ = dF (((0  : Time), v) + ((1 : Time), (0 : X))) := by simp only [Prod.mk_add_mk, zero_add,        add_zero]      _ = dF ((0 : Time), v) + dF ((1 : Time), (0 : X)) := by        simpa using          (dF.map_add ((0 : Time), v) ((1 : Time), (0 : X)))      _ = dF ((0 : Time), v) := by        simp [h_time]      _ = φ v := by        simp [φ]   -- Frechet–Riesz: represent φ as inner product with some g.  refine ⟨(InnerProductSpace.toDual ℝ (X)).symm φ, ?_⟩  intro v  have hinner :      ⟪(InnerProductSpace.toDual ℝ (X)).symm φ, v⟫_ℝ = φ v := by    rw [InnerProductSpace.toDual_symm_apply (𝕜 := ℝ)        (E := X) (x := v) (y := φ)]  calc    δL v = φ v := hφ v    _ = ⟪(InnerProductSpace.toDual ℝ (X)).symm φ, v⟫_ℝ := by      rw [hinner.symm]