-- Lean 4 proof step from Mathlib.RingTheory.Etale.Kaehler (goal)
Goal:
R : Type u_1
S : Type u_2
T : Type u_3
inst : CommRing R
inst_1 : CommRing S
inst_2 : CommRing T
inst_3 : Algebra R S
inst_4 : Algebra R T
inst_5 : Algebra S T
inst_6 : IsScalarTower R S T
inst_7 : Algebra.FormallyEtale S T
⊢ Eq (α := T ⊗[S] Ω[S⁄R] →ₗ[S] Ω[T⁄R])
    (lift
      ((↑S :
          {S_1 M : Type u_3} →
            {M₂ : Type (max u_2 u_3)} →
              [inst : Semiring S] →
                [inst_8 : Semiring S_1] →
                  [inst_9 : AddCommMonoid M] →
                    [inst_10 : AddCommMonoid M₂] →
                      [inst_11 : Module S M] →
                        [inst_12 : Module S M₂] →
                          [inst_13 : Module S_1 M] →
                            [inst_14 : Module S_1 M₂] →
                              [LinearMap.CompatibleSMul M M₂ S S_1] → (fₗ : M →ₗ[S_1] M₂) → M →ₗ[S] M₂)
        (S := T)
        (((LinearMap.flip
                (Algebra.linearMap T (Module.End T (LinearMap.{u_2, u_2, u_2, u_3} (RingHom.id S) Ω[S⁄R] Ω[T⁄R]))) :
              LinearMap.{u_2, u_2, u_2, u_3} (RingHom.id S) Ω[S⁄R] Ω[T⁄R] →ₗ[T]
                T →ₗ[T] LinearMap.{u_2, u_2, u_2, u_3} (RingHom.id S) Ω[S⁄R] Ω[T⁄R]) :
            (a : Ω[S⁄R] →ₗ[S] Ω[T⁄R]) → T →ₗ[T] Ω[S⁄R] →ₗ[S] Ω[T⁄R])
          (map R R S T))))
    ((↑S :
        {S_1 : Type u_3} →
          {M : Type (max u_2 u_3)} →
            {M₂ : Type u_3} →
              [inst : Semiring S] →
                [inst_8 : Semiring S_1] →
                  [inst_9 : AddCommMonoid M] →
                    [inst_10 : AddCommMonoid M₂] →
                      [inst_11 : Module S M] →
                        [inst_12 : Module S M₂] →
                          [inst_13 : Module S_1 M] →
                            [inst_14 : Module S_1 M₂] →
                              [LinearMap.CompatibleSMul M M₂ S S_1] → (fₗ : M →ₗ[S_1] M₂) → M →ₗ[S] M₂)
      (↑(tensorKaehlerEquivOfFormallyEtale R S T) : T ⊗[S] Ω[S⁄R] →ₗ[T] Ω[T⁄R]))
have h : ∀ (x : T) (a : S),
  Eq.{u_3 + 1} (α := Ω[T⁄R])
    (((((LinearMap.compDer
                    (((AlgebraTensorModule.curry
                            (lift
                              ((↑S :
                                  {S_1 M : Type u_3} →
                                    {M₂ : Type (max u_2 u_3)} →
                                      [inst : Semiring S] →
                                        [inst_8 : Semiring S_1] →
                                          [inst_9 : AddCommMonoid M] →
                                            [inst_10 : AddCommMonoid M₂] →
                                              [inst_11 : Module S M] →
                                                [inst_12 : Module S M₂] →
                                                  [inst_13 : Module S_1 M] →
                                                    [inst_14 : Module S_1 M₂] →
                                                      [LinearMap.CompatibleSMul M M₂ S S_1] →
                                                        (fₗ : M →ₗ[S_1] M₂) → M →ₗ[S] M₂)
                                (S := T)
                                (((LinearMap.flip
                                        (Algebra.linearMap T
                                          (Module.End T
                                            (LinearMap.{u_2, u_2, u_2, u_3} (RingHom.id S) Ω[S⁄R] Ω[T⁄R]))) :
                                      LinearMap.{u_2, u_2, u_2, u_3} (RingHom.id S) Ω[S⁄R] Ω[T⁄R] →ₗ[T]
                                        T →ₗ[T] LinearMap.{u_2, u_2, u_2, u_3} (RingHom.id S) Ω[S⁄R] Ω[T⁄R]) :
                                    (a : Ω[S⁄R] →ₗ[S] Ω[T⁄R]) → T →ₗ[T] Ω[S⁄R] →ₗ[S] Ω[T⁄R])
                                  (map R R S T)))) :
                          T →ₗ[S] LinearMap.{u_2, u_2, u_2, u_3} (RingHom.id S) Ω[S⁄R] Ω[T⁄R]) :
                        (a : T) → Ω[S⁄R] →ₗ[S] Ω[T⁄R])
                      x) :
                  Derivation R S Ω[S⁄R] →ₗ[S] Derivation.{u_1, u_2, u_3} R S Ω[T⁄R]) :
                (a : Derivation R S Ω[S⁄R]) → Derivation R S Ω[T⁄R])
              (D R S) :
            Derivation.{u_1, u_2, u_3} R S Ω[T⁄R]) :
          (a : S) → Ω[T⁄R])
        a :
      Ω[T⁄R])
    (((((LinearMap.compDer
                    (((AlgebraTensorModule.curry
                            ((↑S :
                                {S_1 : Type u_3} →
                                  {M : Type (max u_2 u_3)} →
                                    {M₂ : Type u_3} →
                                      [inst : Semiring S] →
                                        [inst_8 : Semiring S_1] →
                                          [inst_9 : AddCommMonoid M] →
                                            [inst_10 : AddCommMonoid M₂] →
                                              [inst_11 : Module S M] →
                                                [inst_12 : Module S M₂] →
                                                  [inst_13 : Module S_1 M] →
                                                    [inst_14 : Module S_1 M₂] →
                                                      [LinearMap.CompatibleSMul M M₂ S S_1] →
                                                        (fₗ : M →ₗ[S_1] M₂) → M →ₗ[S] M₂)
                              (↑(tensorKaehlerEquivOfFormallyEtale R S T) : T ⊗[S] Ω[S⁄R] →ₗ[T] Ω[T⁄R])) :
                          T →ₗ[S] LinearMap.{u_2, u_2, u_2, u_3} (RingHom.id S) Ω[S⁄R] Ω[T⁄R]) :
                        (a : T) → Ω[S⁄R] →ₗ[S] Ω[T⁄R])
                      x) :
                  Derivation R S Ω[S⁄R] →ₗ[S] Derivation.{u_1, u_2, u_3} R S Ω[T⁄R]) :
                (a : Derivation R S Ω[S⁄R]) → Derivation R S Ω[T⁄R])
              (D R S) :
            Derivation.{u_1, u_2, u_3} R S Ω[T⁄R]) :
          (a : S) → Ω[T⁄R])
        a :
      Ω[T⁄R])
Tactic: ext