-- Module: Mathlib.RingTheory.Unramified.Finite
Goal:
R : Type u_1
S : Type u_2
inst : CommRing R
inst_1 : CommRing S
inst_2 : Algebra R S
M : Type u_3
inst_3 : AddCommGroup M
inst_4 : Module R M
inst_5 : Module S M
inst_6 : IsScalarTower R S M
inst_7 : FormallyUnramified R S
inst_8 : EssFiniteType R S
x : M
h :
  (_root_.TensorProduct.lift
          ((↑R :
              {S M : Type u_2} →
                {M₂ : Type u_3} →
                  [inst : Semiring R] →
                    [inst_9 : Semiring S] →
                      [inst_10 : AddCommMonoid M] →
                        [inst_11 : AddCommMonoid M₂] →
                          [inst_12 : Module R M] →
                            [inst_13 : Module R M₂] →
                              [inst_14 : Module S M] →
                                [inst_15 : Module S M₂] →
                                  [LinearMap.CompatibleSMul M M₂ R S] → (fₗ : M →ₗ[S] M₂) → M →ₗ[R] M₂)
            (LinearMap.flip
                ((↑R :
                    {S : Type u_2} →
                      {M : Type u_3} →
                        {M₂ : Type (max u_3 u_2)} →
                          [inst : Semiring R] →
                            [inst_9 : Semiring S] →
                              [inst_10 : AddCommMonoid M] →
                                [inst_11 : AddCommMonoid M₂] →
                                  [inst_12 : Module R M] →
                                    [inst_13 : Module R M₂] →
                                      [inst_14 : Module S M] →
                                        [inst_15 : Module S M₂] →
                                          [LinearMap.CompatibleSMul M M₂ R S] → (fₗ : M →ₗ[S] M₂) → M →ₗ[R] M₂)
                  (LinearMap.flip
                      (AlgHom.toLinearMap (lsmul S S M) : LinearMap.{u_2, u_2, u_2, u_3} (RingHom.id S) S (End S M)) :
                    M →ₗ[S] S →ₗ[S] M)) :
              LinearMap.{u_2, u_2, u_2, u_3} (RingHom.id S) S (M →ₗ[R] M))) :
        (a : S ⊗[R] M) → M)
      (((TensorProduct.AlgebraTensorModule.map LinearMap.id
                (((LinearMap.flip
                        (AlgHom.toLinearMap (lsmul R R M) : LinearMap.{u_1, u_1, u_2, u_3} (RingHom.id R) S (End R M)) :
                      M →ₗ[R] S →ₗ[R] M) :
                    (a : M) → S →ₗ[R] M)
                  x) :
              S ⊗[R] S →ₗ[S] S ⊗[R] M) :
            (a : S ⊗[R] S) → S ⊗[R] M)
          (elem R S) :
        S ⊗[R] M) =
    x
⊢ Eq (α := M)
    (((TensorProduct.AlgebraTensorModule.lift
                (LinearMap.flip
                  ((↑R :
                      {S : Type u_2} →
                        {M : Type u_3} →
                          {M₂ : Type (max u_3 u_2)} →
                            [inst : Semiring R] →
                              [inst_9 : Semiring S] →
                                [inst_10 : AddCommMonoid M] →
                                  [inst_11 : AddCommMonoid M₂] →
                                    [inst_12 : Module R M] →
                                      [inst_13 : Module R M₂] →
                                        [inst_14 : Module S M] →
                                          [inst_15 : Module S M₂] →
                                            [LinearMap.CompatibleSMul M M₂ R S] → (fₗ : M →ₗ[S] M₂) → M →ₗ[R] M₂)
                    (LinearMap.flip
                        (AlgHom.toLinearMap (lsmul S S M) : LinearMap.{u_2, u_2, u_2, u_3} (RingHom.id S) S (End S M)) :
                      M →ₗ[S] S →ₗ[S] M))) :
              S ⊗[R] M →ₗ[S] M) ∘ₗ
            (sec R S M : M →ₗ[S] S ⊗[R] M) :
          (a : M) → M)
        x :
      M)
    ((LinearMap.id : (a : M) → M) x : M)

Tactic:
simp only [sec, LinearMap.coe_comp, LinearMap.coe_mk, LinearMap.coe_toAddHom,
    Function.comp_apply, LinearMap.flip_apply, TensorProduct.AlgebraTensorModule.mapBilinear_apply,
    TensorProduct.AlgebraTensorModule.lift_apply, LinearMap.id_coe, id_eq]

Premises used: LinearMap.coe_mk, LinearMap.id_coe, Algebra.FormallyUnramified.sec, LinearMap.flip_apply, TensorProduct.AlgebraTensorModule.mapBilinear_apply, LinearMap.coe_comp, TensorProduct.AlgebraTensorModule.lift_apply, LinearMap.coe_toAddHom, Function.comp_apply, id_eq