-- Lean 4 proof step from Mathlib.RingTheory.Smooth.Kaehler (hypothesis)
Goal:
R : Type u_1
P : Type u_2
S : Type u_3
inst : CommRing R
inst_1 : CommRing P
inst_2 : CommRing S
inst_3 : Algebra R P
inst_4 : Algebra P S
inst_5 : Algebra R S
inst_6 : IsScalarTower R P S
hf' : HPow.hPow (α := Ideal P) (β := ℕ) (RingHom.ker (F := P →+* S) (algebraMap P S)) 2 = ⊥
hf : Surjective.{u_2 + 1, u_3 + 1} (α := P) (β := S) (⇑(algebraMap P S) : (a : P) → S)
l : { l : S ⊗[P] Ω[P⁄R] →ₗ[P] ↥(RingHom.ker (F := P →+* S) (algebraMap P S)) // l ∘ₗ kerToTensor R P S = LinearMap.id }
p s : P
⊢ Eq.{u_2 + 1} (α := P)
    (↑(((((LinearMap.compDer
                      (((AlgebraTensorModule.curry
                              (↑((fun
                                        (g :
                                          { g : S →ₐ[R] P //
                                            AlgHom.comp (IsScalarTower.toAlgHom R P S) g = AlgHom.id R S }) ↦
                                      (Subtype.mk (p :=
                                          fun (l : S ⊗[P] Ω[P⁄R] →ₗ[P] ↥(RingHom.ker (F := P →+* S) (algebraMap P S))) ↦
                                          l ∘ₗ kerToTensor R P S = LinearMap.id)
                                          (retractionOfSectionOfKerSqZero (↑g : S →ₐ[R] P) hf' ⋯) ⋯ :
                                        { l : S ⊗[P] Ω[P⁄R] →ₗ[P] ↥(RingHom.ker (F := P →+* S) (algebraMap P S)) //
                                          l ∘ₗ kerToTensor R P S = LinearMap.id }))
                                    ((fun
                                          (l :
                                            { l : S ⊗[P] Ω[P⁄R] →ₗ[P] ↥(RingHom.ker (F := P →+* S) (algebraMap P S)) //
                                              l ∘ₗ kerToTensor R P S = LinearMap.id }) ↦
                                        (Subtype.mk (p := fun (g : S →ₐ[R] P) ↦
                                            AlgHom.comp (IsScalarTower.toAlgHom R P S) g = AlgHom.id R S)
                                            (sectionOfRetractionKerToTensor
                                              (↑l : S ⊗[P] Ω[P⁄R] →ₗ[P] ↥(RingHom.ker (algebraMap P S))) ⋯ hf' hf)
                                            ⋯ :
                                          { g : S →ₐ[R] P //
                                            AlgHom.comp (IsScalarTower.toAlgHom R P S) g = AlgHom.id R S }))
                                      l)) :
                                S ⊗[P] Ω[P⁄R] →ₗ[P] ↥(RingHom.ker (algebraMap P S))) :
                            S →ₗ[P] Ω[P⁄R] →ₗ[P] ↥(RingHom.ker (F := P →+* S) (algebraMap P S))) :
                          (a : S) → Ω[P⁄R] →ₗ[P] ↥(RingHom.ker (algebraMap P S)))
                        ((algebraMap P S : (a : P) → S) s)) :
                    Derivation R P Ω[P⁄R] →ₗ[P] Derivation R P ↥(RingHom.ker (F := P →+* S) (algebraMap P S))) :
                  (a : Derivation R P Ω[P⁄R]) → Derivation R P ↥(RingHom.ker (algebraMap P S)))
                (D R P) :
              Derivation R P ↥(RingHom.ker (F := P →+* S) (algebraMap P S))) :
            (a : P) → ↥(RingHom.ker (algebraMap P S)))
          p) :
      P)
    (↑(((((LinearMap.compDer
                      (((AlgebraTensorModule.curry (↑l : S ⊗[P] Ω[P⁄R] →ₗ[P] ↥(RingHom.ker (algebraMap P S))) :
                            S →ₗ[P] Ω[P⁄R] →ₗ[P] ↥(RingHom.ker (F := P →+* S) (algebraMap P S))) :
                          (a : S) → Ω[P⁄R] →ₗ[P] ↥(RingHom.ker (algebraMap P S)))
                        ((algebraMap P S : (a : P) → S) s)) :
                    Derivation R P Ω[P⁄R] →ₗ[P] Derivation R P ↥(RingHom.ker (F := P →+* S) (algebraMap P S))) :
                  (a : Derivation R P Ω[P⁄R]) → Derivation R P ↥(RingHom.ker (algebraMap P S)))
                (D R P) :
              Derivation R P ↥(RingHom.ker (F := P →+* S) (algebraMap P S))) :
            (a : P) → ↥(RingHom.ker (algebraMap P S)))
          p) :
      P)
have this : ∀ (x y : S ⊗[P] Ω[P⁄R]),
  HMul.hMul (α := P)
      (↑(((↑l : S ⊗[P] Ω[P⁄R] →ₗ[P] ↥(RingHom.ker (algebraMap P S))) :
              (a : S ⊗[P] Ω[P⁄R]) → ↥(RingHom.ker (algebraMap P S)))
            x) :
        P)
      (↑(((↑l : S ⊗[P] Ω[P⁄R] →ₗ[P] ↥(RingHom.ker (algebraMap P S))) :
              (a : S ⊗[P] Ω[P⁄R]) → ↥(RingHom.ker (algebraMap P S)))
            y) :
        P) =
    0
Tactic: have (x y : _) : (l.1 x).1 * (l.1 y).1 = 0 := by
      rw [← Ideal.mem_bot, ← hf', pow_two]; exact Ideal.mul_mem_mul (l.1 x).2 (l.1 y).2