-- Module: Mathlib.Analysis.Calculus.VectorField
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
E : Type u_2
inst_1 : NormedAddCommGroup E
inst_2 : NormedSpace 𝕜 E
F : Type u_3
inst_3 : NormedAddCommGroup F
inst_4 : NormedSpace 𝕜 F
inst_5 : CompleteSpace E
f : E → F
s : Set E
x : E
h'f : ContDiffWithinAt 𝕜 2 f s x
hs : UniqueDiffOn 𝕜 s
hx : x ∈ s
M : E ≃L[𝕜] F
hM : (↑M : E →L[𝕜] F) = fderivWithin 𝕜 f s x
U : Set E := {y : E | ∃ (N : E ≃L[𝕜] F), (↑N : E →L[𝕜] F) = fderivWithin 𝕜 f s y}
hU : U ∈ 𝓝[s] x
N : E → E ≃L[𝕜] F :=
  fun (x : E) ↦
    if h : x ∈ U then Exists.choose (p := fun (N : E ≃L[𝕜] F) ↦ (↑N : E →L[𝕜] F) = fderivWithin 𝕜 f s x) h else M
eN : (𝓝[s] x).EventuallyEq (β := E →L[𝕜] F) (fun (y : E) ↦ (↑(N y) : E →L[𝕜] F)) fun (y : E) ↦ fderivWithin 𝕜 f s y
e'N : (↑(N x) : E →L[𝕜] F) = fderivWithin 𝕜 f s x
hN : ContDiffWithinAt 𝕜 (F := E →L[𝕜] F) 1 (fun (y : E) ↦ (↑(N y) : E →L[𝕜] F)) s x
hN' : ContDiffWithinAt 𝕜 (F := F →L[𝕜] E) 1 (fun (y : E) ↦ (↑(N y).symm : F →L[𝕜] E)) s x
v : E
A' :
  ∀ (y : E),
    Eq.{u_3 + 1} (α := F →L[𝕜] F)
      ((((ContinuousLinearMap.compL 𝕜 F E F : (a : E →L[𝕜] F) → (F →L[𝕜] E) →L[𝕜] F →L[𝕜] F) (↑(N y) : E →L[𝕜] F) :
              (F →L[𝕜] E) →L[𝕜] F →L[𝕜] F) :
            (a : F →L[𝕜] E) → F →L[𝕜] F)
          (↑(N y).symm : F →L[𝕜] E) :
        F →L[𝕜] F)
      (ContinuousLinearMap.id 𝕜 F : F →L[𝕜] F)
this :
  Eq.{u_3 + 1} (α := F →L[𝕜] F)
    ((fderivWithin 𝕜
            (fun (y : E) ↦
              (((ContinuousLinearMap.compL 𝕜 F E F : (a : E →L[𝕜] F) → (F →L[𝕜] E) →L[𝕜] F →L[𝕜] F)
                      (↑(N y) : E →L[𝕜] F) :
                    (F →L[𝕜] E) →L[𝕜] F →L[𝕜] F) :
                  (a : F →L[𝕜] E) → F →L[𝕜] F)
                (↑(N y).symm : F →L[𝕜] E))
            s x :
          (a : E) → F →L[𝕜] F)
        v :
      F →L[𝕜] F)
    (0 : F →L[𝕜] F)
⊢ Eq.{u_3 + 1} (α := F →L[𝕜] F)
    ((HAdd.hAdd (α := ContinuousLinearMap.{u_1, u_1, u_2, u_3} (RingHom.id 𝕜) E (F →L[𝕜] F)) (β :=
            ContinuousLinearMap.{u_1, u_1, u_2, u_3} (RingHom.id 𝕜) E (F →L[𝕜] F))
            ((((ContinuousLinearMap.precompR E (ContinuousLinearMap.compL 𝕜 F E F) :
                        (a : E →L[𝕜] F) → (E →L[𝕜] F →L[𝕜] E) →L[𝕜] E →L[𝕜] F →L[𝕜] F)
                      (↑(N x) : E →L[𝕜] F) :
                    (E →L[𝕜] F →L[𝕜] E) →L[𝕜] ContinuousLinearMap.{u_1, u_1, u_2, u_3} (RingHom.id 𝕜) E (F →L[𝕜] F)) :
                  (a : E →L[𝕜] F →L[𝕜] E) → E →L[𝕜] F →L[𝕜] F)
                (fderivWithin 𝕜 (fun (y : E) ↦ (↑(N y).symm : F →L[𝕜] E)) s x) :
              ContinuousLinearMap.{u_1, u_1, u_2, u_3} (RingHom.id 𝕜) E (F →L[𝕜] F))
            ((((ContinuousLinearMap.precompL E (ContinuousLinearMap.compL 𝕜 F E F) :
                        (a : E →L[𝕜] E →L[𝕜] F) → (F →L[𝕜] E) →L[𝕜] E →L[𝕜] F →L[𝕜] F)
                      (fderivWithin 𝕜 (fun (y : E) ↦ (↑(N y) : E →L[𝕜] F)) s x) :
                    (F →L[𝕜] E) →L[𝕜] ContinuousLinearMap.{u_1, u_1, u_2, u_3} (RingHom.id 𝕜) E (F →L[𝕜] F)) :
                  (a : F →L[𝕜] E) → E →L[𝕜] F →L[𝕜] F)
                (↑(N x).symm : F →L[𝕜] E) :
              ContinuousLinearMap.{u_1, u_1, u_2, u_3} (RingHom.id 𝕜) E (F →L[𝕜] F)) :
          (a : E) → F →L[𝕜] F)
        v :
      F →L[𝕜] F)
    (0 : F →L[𝕜] F)

Tactic:
rw [ContinuousLinearMap.fderivWithin_of_bilinear _ (hN.differentiableWithinAt one_ne_zero)
      (hN'.differentiableWithinAt one_ne_zero) (hs x hx)] at this

Premises used: ContinuousLinearMap.fderivWithin_of_bilinear, one_ne_zero, ContDiffWithinAt.differentiableWithinAt