-- Lean 4 proof step from Mathlib.RingTheory.Kaehler.TensorProduct (goal)
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
h : Algebra.IsPushout R S A B
x : TensorProduct.{u_1, u_2, u_3} R S Ω[A⁄R]
a : A
tmul :
  (LinearMap.liftBaseChange S
          ((↑R :
              {S M : Type u_3} →
                {M₂ : Type u_4} →
                  [inst : Semiring R] →
                    [inst_11 : Semiring S] →
                      [inst_12 : AddCommMonoid M] →
                        [inst_13 : AddCommMonoid M₂] →
                          [inst_14 : Module R M] →
                            [inst_15 : Module R M₂] →
                              [inst_16 : Module S M] →
                                [inst_17 : Module S M₂] →
                                  [LinearMap.CompatibleSMul M M₂ R S] → (fₗ : M →ₗ[S] M₂) → M →ₗ[R] M₂)
            (map R S A B)) :
        (a : S ⊗[R] Ω[A⁄R]) → Ω[B⁄S])
      (HSMul.hSMul (α := B) (β := TensorProduct.{u_1, u_2, u_3} R S Ω[A⁄R])
        ((AlgHom.toLinearMap (IsScalarTower.toAlgHom R A B) : (a : A) → B) a : B) 0) =
    HSMul.hSMul (α := B) (β := Ω[B⁄S]) ((AlgHom.toLinearMap (IsScalarTower.toAlgHom R A B) : (a : A) → B) a : B)
      ((LinearMap.liftBaseChange S
            ((↑R :
                {S M : Type u_3} →
                  {M₂ : Type u_4} →
                    [inst : Semiring R] →
                      [inst_11 : Semiring S] →
                        [inst_12 : AddCommMonoid M] →
                          [inst_13 : AddCommMonoid M₂] →
                            [inst_14 : Module R M] →
                              [inst_15 : Module R M₂] →
                                [inst_16 : Module S M] →
                                  [inst_17 : Module S M₂] →
                                    [LinearMap.CompatibleSMul M M₂ R S] → (fₗ : M →ₗ[S] M₂) → M →ₗ[R] M₂)
              (map R S A B)) :
          (a : S ⊗[R] Ω[A⁄R]) → Ω[B⁄S])
        0)
⊢ (LinearMap.liftBaseChange S
          ((↑R :
              {S M : Type u_3} →
                {M₂ : Type u_4} →
                  [inst : Semiring R] →
                    [inst_11 : Semiring S] →
                      [inst_12 : AddCommMonoid M] →
                        [inst_13 : AddCommMonoid M₂] →
                          [inst_14 : Module R M] →
                            [inst_15 : Module R M₂] →
                              [inst_16 : Module S M] →
                                [inst_17 : Module S M₂] →
                                  [LinearMap.CompatibleSMul M M₂ R S] → (fₗ : M →ₗ[S] M₂) → M →ₗ[R] M₂)
            (map R S A B)) :
        (a : S ⊗[R] Ω[A⁄R]) → Ω[B⁄S])
      (HSMul.hSMul (α := B) ((AlgHom.toLinearMap (IsScalarTower.toAlgHom R A B) : (a : A) → B) a : B) x) =
    HSMul.hSMul (α := B) (β := Ω[B⁄S]) ((AlgHom.toLinearMap (IsScalarTower.toAlgHom R A B) : (a : A) → B) a : B)
      ((LinearMap.liftBaseChange S
              ((↑R :
                  {S M : Type u_3} →
                    {M₂ : Type u_4} →
                      [inst : Semiring R] →
                        [inst_11 : Semiring S] →
                          [inst_12 : AddCommMonoid M] →
                            [inst_13 : AddCommMonoid M₂] →
                              [inst_14 : Module R M] →
                                [inst_15 : Module R M₂] →
                                  [inst_16 : Module S M] →
                                    [inst_17 : Module S M₂] →
                                      [LinearMap.CompatibleSMul M M₂ R S] → (fₗ : M →ₗ[S] M₂) → M →ₗ[R] M₂)
                (map R S A B)) :
            (a : S ⊗[R] Ω[A⁄R]) → Ω[B⁄S])
          x :
        Ω[B⁄S])
have tmul₁ : ∀ (x : S) (y : Ω[A⁄R]),
  (LinearMap.liftBaseChange S
          ((↑R :
              {S M : Type u_3} →
                {M₂ : Type u_4} →
                  [inst : Semiring R] →
                    [inst_11 : Semiring S] →
                      [inst_12 : AddCommMonoid M] →
                        [inst_13 : AddCommMonoid M₂] →
                          [inst_14 : Module R M] →
                            [inst_15 : Module R M₂] →
                              [inst_16 : Module S M] →
                                [inst_17 : Module S M₂] →
                                  [LinearMap.CompatibleSMul M M₂ R S] → (fₗ : M →ₗ[S] M₂) → M →ₗ[R] M₂)
            (map R S A B)) :
        (a : S ⊗[R] Ω[A⁄R]) → Ω[B⁄S])
      (HSMul.hSMul (α := B) (β := TensorProduct.{u_1, u_2, u_3} R S Ω[A⁄R])
        ((AlgHom.toLinearMap (IsScalarTower.toAlgHom R A B) : (a : A) → B) a : B) (x ⊗ₜ[R] y)) =
    HSMul.hSMul (α := B) (β := Ω[B⁄S]) ((AlgHom.toLinearMap (IsScalarTower.toAlgHom R A B) : (a : A) → B) a : B)
      ((LinearMap.liftBaseChange S
              ((↑R :
                  {S M : Type u_3} →
                    {M₂ : Type u_4} →
                      [inst : Semiring R] →
                        [inst_11 : Semiring S] →
                          [inst_12 : AddCommMonoid M] →
                            [inst_13 : AddCommMonoid M₂] →
                              [inst_14 : Module R M] →
                                [inst_15 : Module R M₂] →
                                  [inst_16 : Module S M] →
                                    [inst_17 : Module S M₂] →
                                      [LinearMap.CompatibleSMul M M₂ R S] → (fₗ : M →ₗ[S] M₂) → M →ₗ[R] M₂)
                (map R S A B)) :
            (a : S ⊗[R] Ω[A⁄R]) → Ω[B⁄S])
          (x ⊗ₜ[R] y) :
        Ω[B⁄S])
Tactic: induction x
Cleared hypotheses: x