-- Module: Mathlib.Analysis.Calculus.FDeriv.Analytic
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
F : Type v
inst_1 : NormedAddCommGroup F
inst_2 : NormedSpace 𝕜 F
ι : Type u_2
E : ι → Type u_3
inst_3 : (i : ι) → NormedAddCommGroup (E i)
inst_4 : (i : ι) → NormedSpace.{u_1, u_3} 𝕜 (E i)
inst_5 : Fintype ι
inst_6 : DecidableEq ι
fa : ContinuousMultilinearMap 𝕜 E F × ((i : ι) → E i)
f : ContinuousMultilinearMap 𝕜 (fun (x : Option ι) ↦ ContinuousMultilinearMap 𝕜 E F × ((i : ι) → E i)) F :=
  ContinuousLinearMap.continuousMultilinearMapOption (E := E)
    (ContinuousLinearMap.id 𝕜 (ContinuousMultilinearMap 𝕜 E F))
Hf :
  HasStrictFDerivAt
    (fun (x : ContinuousMultilinearMap 𝕜 E F × ((i : ι) → E i)) ↦
      (f : (a : (i : Option ι) → ContinuousMultilinearMap 𝕜 E F × ((i : ι) → E i)) → F)
        ((fun (fx : ContinuousMultilinearMap 𝕜 E F × ((i : ι) → E i)) (x : Option ι) ↦ fx) x))
    ((f.linearDeriv (M₁ := fun (x : Option ι) ↦ ContinuousMultilinearMap 𝕜 E F × ((i : ι) → E i)) fun (x : Option ι) ↦
        fa) ∘SL
      ContinuousLinearMap.pi (φ := fun (i : Option ι) ↦ ContinuousMultilinearMap 𝕜 E F × ((i : ι) → E i))
        fun (x : Option ι) ↦ ContinuousLinearMap.id 𝕜 (ContinuousMultilinearMap 𝕜 E F × ((i : ι) → E i)))
    fa
e_4 :
  Eq (α := AddCommGroup (ContinuousMultilinearMap 𝕜 E F × ((i : ι) → E i))) Prod.instAddCommGroup
    NormedAddCommGroup.toAddCommGroup
e_5 :
  HEq (α := Module 𝕜 (ContinuousMultilinearMap 𝕜 E F × ((i : ι) → E i)))
    (Prod.instModule : Module 𝕜 (ContinuousMultilinearMap 𝕜 E F × ((i : ι) → E i))) (β :=
    Module 𝕜 (ContinuousMultilinearMap 𝕜 E F × ((i : ι) → E i)))
    (NormedSpace.toModule : Module 𝕜 (ContinuousMultilinearMap 𝕜 E F × ((i : ι) → E i)))
e_6 :
  Eq (α := TopologicalSpace (ContinuousMultilinearMap 𝕜 E F × ((i : ι) → E i))) instTopologicalSpaceProd
    UniformSpace.toTopologicalSpace
h :
  ∀ (g : ContinuousMultilinearMap 𝕜 E F),
    Eq.{v + 1} (α := F)
      (((HAdd.hAdd (α := ContinuousMultilinearMap 𝕜 E F × ((i : ι) → E i) →L[𝕜] F)
                  (apply 𝕜 E F fa.2 ∘SL ContinuousLinearMap.fst 𝕜 (ContinuousMultilinearMap 𝕜 E F) ((i : ι) → E i))
                  (fa.1.linearDeriv (M₁ := E) fa.2 ∘SL
                    ContinuousLinearMap.snd 𝕜 (ContinuousMultilinearMap 𝕜 E F) ((i : ι) → E i))).comp
              (σ₁₂ := RingHom.id 𝕜) (ContinuousLinearMap.inl 𝕜 (ContinuousMultilinearMap 𝕜 E F) ((i : ι) → E i)) :
            (a : ContinuousMultilinearMap 𝕜 E F) → F)
          g :
        F)
      ((ContinuousLinearMap.comp (M₂ := ContinuousMultilinearMap 𝕜 E F × ((i : ι) → E i))
              ((f.linearDeriv (M₁ := fun (x : Option ι) ↦ ContinuousMultilinearMap 𝕜 E F × ((i : ι) → E i))
                  fun (x : Option ι) ↦ fa) ∘SL
                ContinuousLinearMap.pi (φ := fun (i : Option ι) ↦ ContinuousMultilinearMap 𝕜 E F × ((i : ι) → E i))
                  fun (x : Option ι) ↦ ContinuousLinearMap.id 𝕜 (ContinuousMultilinearMap 𝕜 E F × ((i : ι) → E i)))
              (ContinuousLinearMap.inl 𝕜 (ContinuousMultilinearMap 𝕜 E F) ((i : ι) → E i)) :
            (a : ContinuousMultilinearMap 𝕜 E F) → F)
          g :
        F)
h₁ :
  ∀ (g : (i : ι) → E i),
    Eq.{v + 1} (α := F)
      ((HAdd.hAdd (α := ContinuousMultilinearMap 𝕜 E F × ((i : ι) → E i) →L[𝕜] F)
                (apply 𝕜 E F fa.2 ∘SL ContinuousLinearMap.fst 𝕜 (ContinuousMultilinearMap 𝕜 E F) ((i : ι) → E i))
                (fa.1.linearDeriv (M₁ := E) fa.2 ∘SL
                  ContinuousLinearMap.snd 𝕜 (ContinuousMultilinearMap 𝕜 E F) ((i : ι) → E i)) ∘SL
              ContinuousLinearMap.inr 𝕜 (ContinuousMultilinearMap 𝕜 E F) ((i : ι) → E i) :
            (a : (i : ι) → E i) → F)
          g :
        F)
      ((ContinuousLinearMap.comp (M₂ := ContinuousMultilinearMap 𝕜 E F × ((i : ι) → E i))
              ((f.linearDeriv (M₁ := fun (x : Option ι) ↦ ContinuousMultilinearMap 𝕜 E F × ((i : ι) → E i))
                  fun (x : Option ι) ↦ fa) ∘SL
                ContinuousLinearMap.pi (φ := fun (i : Option ι) ↦ ContinuousMultilinearMap 𝕜 E F × ((i : ι) → E i))
                  fun (x : Option ι) ↦ ContinuousLinearMap.id 𝕜 (ContinuousMultilinearMap 𝕜 E F × ((i : ι) → E i)))
              (ContinuousLinearMap.inr 𝕜 (ContinuousMultilinearMap 𝕜 E F) ((i : ι) → E i)) :
            (a : (i : ι) → E i) → F)
          g :
        F)
⊢ HAdd.hAdd (α := ContinuousMultilinearMap 𝕜 E F × ((i : ι) → E i) →L[𝕜] F)
      (apply 𝕜 E F fa.2 ∘SL ContinuousLinearMap.fst 𝕜 (ContinuousMultilinearMap 𝕜 E F) ((i : ι) → E i))
      (fa.1.linearDeriv (M₁ := E) fa.2 ∘SL ContinuousLinearMap.snd 𝕜 (ContinuousMultilinearMap 𝕜 E F) ((i : ι) → E i)) =
    (f.linearDeriv (M₁ := fun (x : Option ι) ↦ ContinuousMultilinearMap 𝕜 E F × ((i : ι) → E i)) fun (x : Option ι) ↦
        fa) ∘SL
      ContinuousLinearMap.pi (φ := fun (i : Option ι) ↦ ContinuousMultilinearMap 𝕜 E F × ((i : ι) → E i))
        fun (x : Option ι) ↦ ContinuousLinearMap.id 𝕜 (ContinuousMultilinearMap 𝕜 E F × ((i : ι) → E i))

Tactic:
ext g