-- Module: Mathlib.Topology.VectorBundle.ContinuousAlternatingMap
Goal:
𝕜 : Type u_1
ι : Type u_2
inst : NontriviallyNormedField 𝕜
B : Type u_3
inst_1 : TopologicalSpace B
F₁ : Type u_4
inst_2 : NormedAddCommGroup F₁
inst_3 : NormedSpace 𝕜 F₁
E₁ : B → Type u_5
inst_4 : (x : B) → AddCommGroup (E₁ x)
inst_5 : (x : B) → Module.{u_1, u_5} 𝕜 (E₁ x)
inst_6 : TopologicalSpace (TotalSpace F₁ E₁)
F₂ : Type u_6
inst_7 : NormedAddCommGroup F₂
inst_8 : NormedSpace 𝕜 F₂
E₂ : B → Type u_7
inst_9 : (x : B) → AddCommGroup (E₂ x)
inst_10 : (x : B) → Module.{u_1, u_7} 𝕜 (E₂ x)
inst_11 : TopologicalSpace (TotalSpace F₂ E₂)
inst_12 : (x : B) → TopologicalSpace (E₁ x)
inst_13 : FiberBundle F₁ E₁
inst_14 : (x : B) → TopologicalSpace (E₂ x)
inst_15 : FiberBundle F₂ E₂
e₁ e₁' : Trivialization (B := B) F₁ (Z := TotalSpace F₁ E₁) (TotalSpace.proj (E := E₁))
e₂ e₂' : Trivialization (B := B) F₂ (Z := TotalSpace F₂ E₂) (TotalSpace.proj (E := E₂))
inst_16 : Trivialization.IsLinear 𝕜 (E := E₁) e₁
inst_17 : Trivialization.IsLinear 𝕜 (E := E₁) e₁'
inst_18 : Trivialization.IsLinear 𝕜 (E := E₂) e₂
inst_19 : Trivialization.IsLinear 𝕜 (E := E₂) e₂'
inst_20 : ∀ (x : B), ContinuousAdd.{u_7} (E₂ x)
inst_21 : ∀ (x : B), ContinuousSMul.{u_1, u_7} 𝕜 (E₂ x)
x : B
h :
  x ∈
    baseSet (F := F₁ [⋀^ι]→L[𝕜] F₂) (Z :=
      TotalSpace (F₁ [⋀^ι]→L[𝕜] F₂) fun (x : B) ↦ ContinuousAlternatingMap.{u_1, u_5, u_7, u_2} 𝕜 (E₁ x) (E₂ x) ι)
      (continuousAlternatingMap 𝕜 ι (E₁ := E₁) (E₂ := E₂) e₁ e₂)
L L' : ContinuousAlternatingMap.{u_1, u_5, u_7, u_2} 𝕜 (E₁ x) (E₂ x) ι
H :
  ∀ (x_1 : ι → F₁),
    Eq.{u_6 + 1} (α := F₂)
      ((((↑(continuousAlternatingMap 𝕜 ι (E₁ := E₁) (E₂ := E₂) e₁ e₂) :
                  (TotalSpace (F₁ [⋀^ι]→L[𝕜] F₂) fun (x : B) ↦ E₁ x [⋀^ι]→L[𝕜] E₂ x) → B × F₁ [⋀^ι]→L[𝕜] F₂)
                (TotalSpace.mk (E := fun (x : B) ↦ ContinuousAlternatingMap.{u_1, u_5, u_7, u_2} 𝕜 (E₁ x) (E₂ x) ι) x
                  (L + L'))).2 :
            (a : ι → F₁) → F₂)
          x_1 :
        F₂)
      ((HAdd.hAdd (α := F₁ [⋀^ι]→L[𝕜] F₂)
              ((↑(continuousAlternatingMap 𝕜 ι (E₁ := E₁) (E₂ := E₂) e₁ e₂) :
                    (TotalSpace (F₁ [⋀^ι]→L[𝕜] F₂) fun (x : B) ↦ E₁ x [⋀^ι]→L[𝕜] E₂ x) → B × F₁ [⋀^ι]→L[𝕜] F₂)
                  (TotalSpace.mk (E := fun (x : B) ↦ ContinuousAlternatingMap.{u_1, u_5, u_7, u_2} 𝕜 (E₁ x) (E₂ x) ι) x
                    L)).2
              ((↑(continuousAlternatingMap 𝕜 ι (E₁ := E₁) (E₂ := E₂) e₁ e₂) :
                    (TotalSpace (F₁ [⋀^ι]→L[𝕜] F₂) fun (x : B) ↦ E₁ x [⋀^ι]→L[𝕜] E₂ x) → B × F₁ [⋀^ι]→L[𝕜] F₂)
                  (TotalSpace.mk (E := fun (x : B) ↦ ContinuousAlternatingMap.{u_1, u_5, u_7, u_2} 𝕜 (E₁ x) (E₂ x) ι) x
                    L')).2 :
            (a : ι → F₁) → F₂)
          x_1 :
        F₂)
⊢ ((↑(continuousAlternatingMap 𝕜 ι (E₁ := E₁) (E₂ := E₂) e₁ e₂) :
          (TotalSpace (F₁ [⋀^ι]→L[𝕜] F₂) fun (x : B) ↦ E₁ x [⋀^ι]→L[𝕜] E₂ x) → B × F₁ [⋀^ι]→L[𝕜] F₂)
        (TotalSpace.mk (E := fun (x : B) ↦ ContinuousAlternatingMap.{u_1, u_5, u_7, u_2} 𝕜 (E₁ x) (E₂ x) ι) x
          (L + L'))).2 =
    HAdd.hAdd (α := F₁ [⋀^ι]→L[𝕜] F₂)
      ((↑(continuousAlternatingMap 𝕜 ι (E₁ := E₁) (E₂ := E₂) e₁ e₂) :
            (TotalSpace (F₁ [⋀^ι]→L[𝕜] F₂) fun (x : B) ↦ E₁ x [⋀^ι]→L[𝕜] E₂ x) → B × F₁ [⋀^ι]→L[𝕜] F₂)
          (TotalSpace.mk (E := fun (x : B) ↦ ContinuousAlternatingMap.{u_1, u_5, u_7, u_2} 𝕜 (E₁ x) (E₂ x) ι) x L)).2
      ((↑(continuousAlternatingMap 𝕜 ι (E₁ := E₁) (E₂ := E₂) e₁ e₂) :
            (TotalSpace (F₁ [⋀^ι]→L[𝕜] F₂) fun (x : B) ↦ E₁ x [⋀^ι]→L[𝕜] E₂ x) → B × F₁ [⋀^ι]→L[𝕜] F₂)
          (TotalSpace.mk (E := fun (x : B) ↦ ContinuousAlternatingMap.{u_1, u_5, u_7, u_2} 𝕜 (E₁ x) (E₂ x) ι) x L')).2

Tactic:
ext