-- Module: Mathlib.RingTheory.Kaehler.JacobiZariski
Goal:
R : Type u₁
S : Type u₂
inst : CommRing R
inst_1 : CommRing S
inst_2 : Algebra R S
T : Type u₃
inst_3 : CommRing T
inst_4 : Algebra R T
inst_5 : Algebra S T
inst_6 : IsScalarTower R S T
ι : Type w₁
σ : Type w₂
Q : Generators S T ι
P : Generators R S σ
x : Generators.Ring (Q.comp P)
hx' : Membership.mem (γ := Ideal (Extension.Ring (toExtension (Q.comp P)))) (Extension.ker (toExtension (Q.comp P))) x
y : Generators.Ring (Q.comp P)
hy : y ∈ HMul.hMul (α := Ideal (Generators.Ring (Q.comp P))) (ker (Q.comp P)) (ker (Q.comp P))
e :
  Membership.mem (γ := Ideal (Generators.Ring (Q.comp P)))
    (Ideal.map (F := P.Ring →ₐ[R] Generators.Ring (Q.comp P)) (Hom.toAlgHom (Q.toComp P)) P.ker) (x - y)
z : ↥(ker (Q.comp P)) :=
  Subtype.mk (p := fun (x : Generators.Ring (Q.comp P)) ↦
    Membership.mem (γ := Ideal (Generators.Ring (Q.comp P))) (ker (Q.comp P)) x) (x - y) ⋯
hz :
  Membership.mem (γ := Ideal (Generators.Ring (Q.comp P)))
    (Ideal.map (F := P.Ring →+* Generators.Ring (Q.comp P)) (AlgHom.toRingHom (Hom.toAlgHom (Q.toComp P))) P.ker)
    (↑z : (Q.comp P).Ring)
this :
  Eq (α := Extension.Cotangent (toExtension (Q.comp P)))
    ((Extension.Cotangent.mk : (a : ↥(Q.comp P).toExtension.ker) → (Q.comp P).toExtension.Cotangent)
        (Subtype.mk (p := fun (x : Extension.Ring (toExtension (Q.comp P))) ↦
          Membership.mem (γ := Ideal (Extension.Ring (toExtension (Q.comp P)))) (Extension.ker (toExtension (Q.comp P)))
            x)
          x hx') :
      Extension.Cotangent (toExtension (Q.comp P)))
    ((Extension.Cotangent.mk : (a : ↥(Q.comp P).toExtension.ker) → (Q.comp P).toExtension.Cotangent) z :
      Extension.Cotangent (toExtension (Q.comp P)))
w : P.Ring
hw : w ∈ P.ker
h2 :
  Membership.mem (α := Generators.Ring (Q.comp P)) (γ :=
    Submodule (Generators.Ring (Q.comp P)) (Generators.Ring (Q.comp P)))
    (Extension.ker (toExtension (Q.comp P)) : Ideal (Extension.Ring (toExtension (Q.comp P))))
    ((Hom.toAlgHom (Q.toComp P) : (a : P.Ring) → (Q.comp P).Ring) w : Generators.Ring (Q.comp P))
h2₁ :
  Eq (α := Extension.Cotangent (toExtension (Q.comp P)))
    ((Extension.Cotangent.map (Hom.toExtensionHom (Q.toComp P)) :
          (a : P.toExtension.Cotangent) → (Q.comp P).toExtension.Cotangent)
        ((Extension.Cotangent.mk : (a : ↥P.toExtension.ker) → P.toExtension.Cotangent)
          (Subtype.mk (p := fun (x : P.toExtension.Ring) ↦ x ∈ P.toExtension.ker) w hw)) :
      Extension.Cotangent (toExtension (Q.comp P)))
    (((Extension.Cotangent.mk :
            LinearMap (R := Extension.Ring (toExtension (Q.comp P)))
              (RingHom.id (Extension.Ring (toExtension (Q.comp P)))) (↥(Extension.ker (toExtension (Q.comp P))))
              (Extension.Cotangent (toExtension (Q.comp P)))) :
          (a : ↥(Q.comp P).toExtension.ker) → (Q.comp P).toExtension.Cotangent)
        (Subtype.mk (p := fun (a : Generators.Ring (Q.comp P)) ↦
          Membership.mem (γ := Submodule (Generators.Ring (Q.comp P)) (Generators.Ring (Q.comp P)))
            (Extension.ker (toExtension (Q.comp P))) a)
          ((Hom.toAlgHom (Q.toComp P) : (a : P.Ring) → (Q.comp P).Ring) w) ⋯) :
      Extension.Cotangent (toExtension (Q.comp P)))
⊢ ∃ (x :
    Membership.mem (α := Generators.Ring (Q.comp P)) (γ :=
      Submodule (Generators.Ring (Q.comp P)) (Generators.Ring (Q.comp P)))
      (Extension.ker (toExtension (Q.comp P)) : Ideal (Extension.Ring (toExtension (Q.comp P))))
      ((Hom.toAlgHom (Q.toComp P) : (a : P.Ring) → (Q.comp P).Ring) w : Generators.Ring (Q.comp P))),
    Membership.mem (α := Extension.Cotangent (toExtension (Q.comp P))) (γ :=
      Submodule T (Extension.Cotangent (toExtension (Q.comp P))))
      (Submodule.span T
        (↑(LinearMap.range (Extension.Cotangent.map (Hom.toExtensionHom (Q.toComp P)))) :
          Set (Q.comp P).toExtension.Cotangent))
      (((Extension.Cotangent.mk :
              LinearMap (R := Extension.Ring (toExtension (Q.comp P)))
                (RingHom.id (Extension.Ring (toExtension (Q.comp P)))) (↥(Extension.ker (toExtension (Q.comp P))))
                (Extension.Cotangent (toExtension (Q.comp P)))) :
            (a : ↥(Q.comp P).toExtension.ker) → (Q.comp P).toExtension.Cotangent)
          (Subtype.mk (p := fun (a : Generators.Ring (Q.comp P)) ↦
            Membership.mem (γ := Submodule (Generators.Ring (Q.comp P)) (Generators.Ring (Q.comp P)))
              (Extension.ker (toExtension (Q.comp P))) a)
            ((Hom.toAlgHom (Q.toComp P) : (a : P.Ring) → (Q.comp P).Ring) w) ⋯) :
        Extension.Cotangent (toExtension (Q.comp P)))

Tactic:
refine ⟨?_, Submodule.subset_span ⟨Extension.Cotangent.mk ⟨w, hw⟩, ?_⟩⟩