-- Module: Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno
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
G : Type u_4
inst_5 : NormedAddCommGroup G
inst_6 : NormedSpace 𝕜 G
m : ℕ
q : FormalMultilinearSeries 𝕜 F G
p : FormalMultilinearSeries 𝕜 E F
c : OrderedFinpartition m
h :
  ∀ (e : E) (v : (i : Fin m) → E),
    Eq.{u_4 + 1} (α := G)
      (((((q.compAlongOrderedFinpartition p (c.extend none)).curryLeft (Ei := fun (i : Fin (m + 1)) ↦ E) :
                  (a : E) → ContinuousMultilinearMap 𝕜 (fun (i : Fin m) ↦ E) G)
                e :
              ContinuousMultilinearMap.{u_1, 0, u_2, u_4} 𝕜 (fun (i : Fin m) ↦ E) G) :
            (a : (i : Fin m) → E) → G)
          v :
        G)
      ((((ContinuousLinearMap.comp (σ₂₃ := RingHom.id 𝕜) (M₂ :=
                    ContinuousMultilinearMap.{u_1, 0, u_3, u_4} 𝕜 (fun (i : Fin c.length) ↦ F) G)
                    ((ContinuousLinearMap.flipMultilinear (E := fun (i : Fin c.length) ↦
                          ContinuousMultilinearMap.{u_1, 0, u_2, u_3} 𝕜 (fun (i : Fin (c.partSize i)) ↦ E) F)
                          (compAlongOrderedFinpartitionL 𝕜 E F G c) :
                        (a : (i : Fin c.length) → ContinuousMultilinearMap 𝕜 (fun (i : Fin (c.partSize i)) ↦ E) F) →
                          ContinuousMultilinearMap 𝕜 (fun (i : Fin c.length) ↦ F) G →L[𝕜]
                            ContinuousMultilinearMap 𝕜 (fun (i : Fin m) ↦ E) G)
                      fun (i : Fin c.length) ↦ p (c.partSize i))
                    (ContinuousLinearMap.comp (σ₁₂ := RingHom.id 𝕜)
                      (ContinuousMultilinearMap.curryLeft (Ei := fun (i : Fin (c.length + 1)) ↦ F) (q (c.length + 1)))
                      ((continuousMultilinearCurryFin1 𝕜 E F :
                          (a : ContinuousMultilinearMap 𝕜 (fun (i : Fin 1) ↦ E) F) → E →L[𝕜] F)
                        (p 1))) :
                  (a : E) → ContinuousMultilinearMap 𝕜 (fun (i : Fin m) ↦ E) G)
                e :
              ContinuousMultilinearMap.{u_1, 0, u_2, u_4} 𝕜 (fun (i : Fin m) ↦ E) G) :
            (a : (i : Fin m) → E) → G)
          v :
        G)
⊢ (q.compAlongOrderedFinpartition p (c.extend none)).curryLeft (Ei := fun (i : Fin (m + 1)) ↦ E) =
    ContinuousLinearMap.comp (σ₂₃ := RingHom.id 𝕜) (M₂ :=
      ContinuousMultilinearMap.{u_1, 0, u_3, u_4} 𝕜 (fun (i : Fin c.length) ↦ F) G)
      ((ContinuousLinearMap.flipMultilinear (E := fun (i : Fin c.length) ↦
            ContinuousMultilinearMap.{u_1, 0, u_2, u_3} 𝕜 (fun (i : Fin (c.partSize i)) ↦ E) F)
            (compAlongOrderedFinpartitionL 𝕜 E F G c) :
          (a : (i : Fin c.length) → ContinuousMultilinearMap 𝕜 (fun (i : Fin (c.partSize i)) ↦ E) F) →
            ContinuousMultilinearMap 𝕜 (fun (i : Fin c.length) ↦ F) G →L[𝕜]
              ContinuousMultilinearMap 𝕜 (fun (i : Fin m) ↦ E) G)
        fun (i : Fin c.length) ↦ p (c.partSize i))
      (ContinuousLinearMap.comp (σ₁₂ := RingHom.id 𝕜)
        (ContinuousMultilinearMap.curryLeft (Ei := fun (i : Fin (c.length + 1)) ↦ F) (q (c.length + 1)))
        ((continuousMultilinearCurryFin1 𝕜 E F : (a : ContinuousMultilinearMap 𝕜 (fun (i : Fin 1) ↦ E) F) → E →L[𝕜] F)
          (p 1)))

Tactic:
ext e v