-- Module: Mathlib.RingTheory.Kaehler.TensorProduct
Goal:
R : Type u_1
S : Type u_2
A : Type u_3
B : Type u_4
inst : CommRing R
inst_1 : CommRing S
inst_2 : Algebra R S
inst_3 : CommRing A
inst_4 : CommRing B
inst_5 : Algebra R A
inst_6 : Algebra R B
inst_7 : Algebra A B
inst_8 : Algebra S B
inst_9 : IsScalarTower R A B
inst_10 : IsScalarTower R S B
inst_11 : Algebra.IsPushout R S A B
H :
  ∀ (x : S) (y : Ω[A⁄R]),
    Eq (α := TensorProduct.{u_1, u_2, u_3} R S Ω[A⁄R])
      ((((↑R :
                  {S : Type u_2} →
                    {M M₂ : Type (max u_2 u_3)} →
                      [inst : Semiring R] →
                        [inst_12 : Semiring S] →
                          [inst_13 : AddCommMonoid M] →
                            [inst_14 : AddCommMonoid M₂] →
                              [inst_15 : Module R M] →
                                [inst_16 : Module R M₂] →
                                  [inst_17 : Module S M] →
                                    [inst_18 : Module S M₂] →
                                      [LinearMap.CompatibleSMul M M₂ R S] → (fₗ : M →ₗ[S] M₂) → M →ₗ[R] M₂)
                ((↑S :
                      {S_1 M : Type u_4} →
                        {M₂ : Type (max u_2 u_3)} →
                          [inst : Semiring S] →
                            [inst_12 : Semiring S_1] →
                              [inst_13 : AddCommMonoid M] →
                                [inst_14 : AddCommMonoid M₂] →
                                  [inst_15 : Module S M] →
                                    [inst_16 : Module S M₂] →
                                      [inst_17 : Module S_1 M] →
                                        [inst_18 : Module S_1 M₂] →
                                          [LinearMap.CompatibleSMul M M₂ S S_1] → (fₗ : M →ₗ[S_1] M₂) → M →ₗ[S] M₂)
                    (Derivation.liftKaehlerDifferential (R := S) (S := B) (M :=
                      TensorProduct.{u_1, u_2, u_3} R S Ω[A⁄R]) (derivationTensorProduct R S A B)) ∘ₗ
                  LinearMap.liftBaseChange S
                    ((↑R :
                        {S M : Type u_3} →
                          {M₂ : Type u_4} →
                            [inst : Semiring R] →
                              [inst_12 : Semiring S] →
                                [inst_13 : AddCommMonoid M] →
                                  [inst_14 : AddCommMonoid M₂] →
                                    [inst_15 : Module R M] →
                                      [inst_16 : Module R M₂] →
                                        [inst_17 : Module S M] →
                                          [inst_18 : Module S M₂] →
                                            [LinearMap.CompatibleSMul M M₂ R S] → (fₗ : M →ₗ[S] M₂) → M →ₗ[R] M₂)
                      (map R S A B))) :
              TensorProduct.{u_1, u_2, u_3} R S Ω[A⁄R] →ₗ[R] TensorProduct.{u_1, u_2, u_3} R S Ω[A⁄R]) :
            (a : S ⊗[R] Ω[A⁄R]) → S ⊗[R] Ω[A⁄R])
          (x ⊗ₜ[R] y) :
        TensorProduct.{u_1, u_2, u_3} R S Ω[A⁄R])
      (((↑R :
                {S : Type u_2} →
                  {M M₂ : Type (max u_2 u_3)} →
                    [inst : Semiring R] →
                      [inst_12 : Semiring S] →
                        [inst_13 : AddCommMonoid M] →
                          [inst_14 : AddCommMonoid M₂] →
                            [inst_15 : Module R M] →
                              [inst_16 : Module R M₂] →
                                [inst_17 : Module S M] →
                                  [inst_18 : Module S M₂] →
                                    [LinearMap.CompatibleSMul M M₂ R S] → (fₗ : M →ₗ[S] M₂) → M →ₗ[R] M₂)
              (S := S) LinearMap.id :
            (a : S ⊗[R] Ω[A⁄R]) → S ⊗[R] Ω[A⁄R])
          (x ⊗ₜ[R] y) :
        TensorProduct.{u_1, u_2, u_3} R S Ω[A⁄R])
⊢ Eq (α := TensorProduct.{u_1, u_2, u_3} R S Ω[A⁄R] →ₗ[R] TensorProduct.{u_1, u_2, u_3} R S Ω[A⁄R])
    ((↑R :
          {S : Type u_2} →
            {M M₂ : Type (max u_2 u_3)} →
              [inst : Semiring R] →
                [inst_12 : Semiring S] →
                  [inst_13 : AddCommMonoid M] →
                    [inst_14 : AddCommMonoid M₂] →
                      [inst_15 : Module R M] →
                        [inst_16 : Module R M₂] →
                          [inst_17 : Module S M] →
                            [inst_18 : Module S M₂] →
                              [LinearMap.CompatibleSMul M M₂ R S] → (fₗ : M →ₗ[S] M₂) → M →ₗ[R] M₂)
        ((↑S :
              {S_1 M : Type u_4} →
                {M₂ : Type (max u_2 u_3)} →
                  [inst : Semiring S] →
                    [inst_12 : Semiring S_1] →
                      [inst_13 : AddCommMonoid M] →
                        [inst_14 : AddCommMonoid M₂] →
                          [inst_15 : Module S M] →
                            [inst_16 : Module S M₂] →
                              [inst_17 : Module S_1 M] →
                                [inst_18 : Module S_1 M₂] →
                                  [LinearMap.CompatibleSMul M M₂ S S_1] → (fₗ : M →ₗ[S_1] M₂) → M →ₗ[S] M₂)
            (Derivation.liftKaehlerDifferential (R := S) (S := B) (M := TensorProduct.{u_1, u_2, u_3} R S Ω[A⁄R])
              (derivationTensorProduct R S A B)) ∘ₗ
          LinearMap.liftBaseChange S
            ((↑R :
                {S M : Type u_3} →
                  {M₂ : Type u_4} →
                    [inst : Semiring R] →
                      [inst_12 : Semiring S] →
                        [inst_13 : AddCommMonoid M] →
                          [inst_14 : AddCommMonoid M₂] →
                            [inst_15 : Module R M] →
                              [inst_16 : Module R M₂] →
                                [inst_17 : Module S M] →
                                  [inst_18 : Module S M₂] →
                                    [LinearMap.CompatibleSMul M M₂ R S] → (fₗ : M →ₗ[S] M₂) → M →ₗ[R] M₂)
              (map R S A B))) :
      TensorProduct.{u_1, u_2, u_3} R S Ω[A⁄R] →ₗ[R] TensorProduct.{u_1, u_2, u_3} R S Ω[A⁄R])
    ((↑R :
          {S : Type u_2} →
            {M M₂ : Type (max u_2 u_3)} →
              [inst : Semiring R] →
                [inst_12 : Semiring S] →
                  [inst_13 : AddCommMonoid M] →
                    [inst_14 : AddCommMonoid M₂] →
                      [inst_15 : Module R M] →
                        [inst_16 : Module R M₂] →
                          [inst_17 : Module S M] →
                            [inst_18 : Module S M₂] →
                              [LinearMap.CompatibleSMul M M₂ R S] → (fₗ : M →ₗ[S] M₂) → M →ₗ[R] M₂)
        (S := S) LinearMap.id :
      TensorProduct.{u_1, u_2, u_3} R S Ω[A⁄R] →ₗ[R] TensorProduct.{u_1, u_2, u_3} R S Ω[A⁄R])

Tactic:
apply TensorProduct.ext'

Premises used: TensorProduct.ext'