-- Lean 4 proof step from Mathlib.Geometry.Manifold.VectorBundle.Hom (goal)
Goal:
𝕜 : Type u_1
F₁ : Type u_2
F₂ : Type u_3
B₁ : Type u_4
B₂ : Type u_5
M : Type u_6
E₁ : B₁ → Type u_7
E₂ : B₂ → Type u_8
inst : NontriviallyNormedField 𝕜
inst_1 : (x : B₁) → AddCommGroup (E₁ x)
inst_2 : (x : B₁) → Module.{u_1, u_7} 𝕜 (E₁ x)
inst_3 : NormedAddCommGroup F₁
inst_4 : NormedSpace 𝕜 F₁
inst_5 : TopologicalSpace (TotalSpace F₁ E₁)
inst_6 : (x : B₁) → TopologicalSpace (E₁ x)
inst_7 : (x : B₂) → AddCommGroup (E₂ x)
inst_8 : (x : B₂) → Module.{u_1, u_8} 𝕜 (E₂ x)
inst_9 : NormedAddCommGroup F₂
inst_10 : NormedSpace 𝕜 F₂
inst_11 : TopologicalSpace (TotalSpace F₂ E₂)
inst_12 : (x : B₂) → TopologicalSpace (E₂ x)
EB₁ : Type u_9
inst_13 : NormedAddCommGroup EB₁
inst_14 : NormedSpace 𝕜 EB₁
HB₁ : Type u_10
inst_15 : TopologicalSpace HB₁
IB₁ : ModelWithCorners 𝕜 EB₁ HB₁
inst_16 : TopologicalSpace B₁
inst_17 : ChartedSpace HB₁ B₁
EB₂ : Type u_11
inst_18 : NormedAddCommGroup EB₂
inst_19 : NormedSpace 𝕜 EB₂
HB₂ : Type u_12
inst_20 : TopologicalSpace HB₂
IB₂ : ModelWithCorners 𝕜 EB₂ HB₂
inst_21 : TopologicalSpace B₂
inst_22 : ChartedSpace HB₂ B₂
EM : Type u_13
inst_23 : NormedAddCommGroup EM
inst_24 : NormedSpace 𝕜 EM
HM : Type u_14
inst_25 : TopologicalSpace HM
IM : ModelWithCorners 𝕜 EM HM
inst_26 : TopologicalSpace M
inst_27 : ChartedSpace HM M
n : WithTop ℕ∞
inst_28 : FiberBundle F₁ E₁
inst_29 : VectorBundle 𝕜 F₁ E₁
inst_30 : FiberBundle F₂ E₂
inst_31 : VectorBundle 𝕜 F₂ E₂
b₁ : M → B₁
b₂ : M → B₂
m₀ : M
ϕ : (m : M) → ContinuousLinearMap.{u_1, u_1, u_7, u_8} (RingHom.id 𝕜) (E₁ (b₁ m)) (E₂ (b₂ m))
v : (m : M) → E₁ (b₁ m)
s : Set M
hϕ :
  ContMDiffWithinAt IM (E' := F₁ →L[𝕜] F₂) 𝓘(𝕜, F₁ →L[𝕜] F₂) (M' := F₁ →L[𝕜] F₂) n
    (fun (m : M) ↦ inCoordinates F₁ E₁ F₂ E₂ (b₁ m₀) (b₁ m) (b₂ m₀) (b₂ m) (ϕ m)) (insert m₀ s) m₀
hv :
  ContMDiffWithinAt.{u_1, u_13, u_14, u_6, u_9, u_10, u_4} IM IB₁ (M' := B₁) n
      (fun (x : M) ↦ TotalSpace.proj.{u_4, u_2, u_7} (F := F₁) (E := E₁) ⟨b₁ x, v x⟩) (insert m₀ s) m₀ ∧
    ContMDiffWithinAt IM 𝓘(𝕜, F₁) (M' := F₁) n
      (fun (x : M) ↦
        ((↑(trivializationAt F₁ E₁ (TotalSpace.proj.{u_4, u_2, u_7} (F := F₁) (E := E₁) ⟨b₁ m₀, v m₀⟩)) :
              TotalSpace F₁ E₁ → B₁ × F₁)
            ⟨b₁ x, v x⟩).2)
      (insert m₀ s) m₀
hb₂ : ContMDiffWithinAt IM IB₂ n b₂ (insert m₀ s) m₀
A : ∀ᶠ (m : M) in 𝓝[insert m₀ s] m₀, b₁ m ∈ Trivialization.baseSet (trivializationAt F₁ E₁ (b₁ m₀))
A' : ∀ᶠ (m : M) in 𝓝[insert m₀ s] m₀, b₂ m ∈ Trivialization.baseSet (trivializationAt F₂ E₂ (b₂ m₀))
m : M
hm : b₁ m ∈ Trivialization.baseSet (trivializationAt F₁ E₁ (b₁ m₀))
h'm : b₂ m ∈ Trivialization.baseSet (trivializationAt F₂ E₂ (b₂ m₀))
⊢ ((↑(trivializationAt F₂ E₂ (b₂ m₀)) : TotalSpace F₂ E₂ → B₂ × F₂)
        ⟨b₂ m, (ϕ m : (a : E₁ (b₁ m)) → E₂ (b₂ m)) (v m)⟩).2 =
    (inCoordinates F₁ E₁ F₂ E₂ (b₁ m₀) (b₁ m) (b₂ m₀) (b₂ m) (ϕ m) : (a : F₁) → F₂)
      ((↑(trivializationAt F₁ E₁ (b₁ m₀)) : TotalSpace F₁ E₁ → B₁ × F₁) ⟨b₁ m, v m⟩).2
have h : ((↑(trivializationAt F₂ E₂ (b₂ m₀)) : TotalSpace F₂ E₂ → B₂ × F₂) ⟨b₂ m, (ϕ m : (a : E₁ (b₁ m)) → E₂ (b₂ m)) (v m)⟩).2 =
  (ContinuousLinearMap.comp (σ₁₂ := RingHom.id 𝕜)
        (↑(Trivialization.continuousLinearEquivAt 𝕜 (E := E₂) (trivializationAt F₂ E₂ (b₂ m₀)) (b₂ m) h'm) :
          E₂ (b₂ m) →L[𝕜] F₂)
        ((ϕ m).comp (σ₁₂ := RingHom.id 𝕜)
          (↑(ContinuousLinearEquiv.symm
                (Trivialization.continuousLinearEquivAt 𝕜 (E := E₁) (trivializationAt F₁ E₁ (b₁ m₀)) (b₁ m) hm)) :
            F₁ →L[𝕜] E₁ (b₁ m))) :
      (a : F₁) → F₂)
    ((↑(trivializationAt F₁ E₁ (b₁ m₀)) : TotalSpace F₁ E₁ → B₁ × F₁) ⟨b₁ m, v m⟩).2
Tactic: rw [inCoordinates_eq hm h'm]