-- Module: Mathlib.RingTheory.Ideal.CotangentBaseChange
Goal:
R : Type u_1
S : Type u_2
inst : CommRing R
inst_1 : CommRing S
inst_2 : Algebra R S
T : Type u_3
inst_3 : CommRing T
inst_4 : Algebra R T
I : Ideal S
a : S →+* T ⊗[R] S := AlgHom.toRingHom Algebra.TensorProduct.includeRight
x : T ⊗[R] S
hx :
  Membership.mem (γ := Ideal (T ⊗[R] S))
    (map (F := S →+* T ⊗[R] S) (AlgHom.toRingHom Algebra.TensorProduct.includeRight) I) x
obtain :
  ∀ (y : TensorProduct.{u_1, u_3, u_2} R T ↥(Submodule.restrictScalars R I))
    (hx :
      Membership.mem (α := T ⊗[R] S) (γ := Ideal (T ⊗[R] S))
        (map (F := S →+* T ⊗[R] S) (AlgHom.toRingHom Algebra.TensorProduct.includeRight) I)
        ((LinearMap.lTensor T (Submodule.subtype (Submodule.restrictScalars R I)) :
              (a : T ⊗[R] ↥(Submodule.restrictScalars R I)) → T ⊗[R] S)
            y :
          T ⊗[R] S)),
    ∃ (a : TensorProduct.{u_1, u_3, u_2} R T I.Cotangent),
      Eq (α :=
        Cotangent (R := T ⊗[R] S) (map (F := S →+* T ⊗[R] S) (AlgHom.toRingHom Algebra.TensorProduct.includeRight) I))
        ((tensorCotangentHom R T I :
              (a : T ⊗[R] I.Cotangent) → (map Algebra.TensorProduct.includeRight.toRingHom I).Cotangent)
            a :
          Cotangent (R := T ⊗[R] S) (map (F := S →+* T ⊗[R] S) (AlgHom.toRingHom Algebra.TensorProduct.includeRight) I))
        ((toCotangent (map (F := S →+* T ⊗[R] S) (AlgHom.toRingHom Algebra.TensorProduct.includeRight) I) :
              (a : ↥(map Algebra.TensorProduct.includeRight.toRingHom I)) →
                (map Algebra.TensorProduct.includeRight.toRingHom I).Cotangent)
            (Subtype.mk (p := fun (x : T ⊗[R] S) ↦
              Membership.mem (γ := Ideal (T ⊗[R] S))
                (map (F := S →+* T ⊗[R] S) (AlgHom.toRingHom Algebra.TensorProduct.includeRight) I) x)
              ((LinearMap.lTensor T (Submodule.subtype (Submodule.restrictScalars R I)) :
                  (a : T ⊗[R] ↥(Submodule.restrictScalars R I)) → T ⊗[R] S)
                y)
              hx) :
          Cotangent (R := T ⊗[R] S) (map (F := S →+* T ⊗[R] S) (AlgHom.toRingHom Algebra.TensorProduct.includeRight) I))
⊢ ∃ (a : TensorProduct.{u_1, u_3, u_2} R T I.Cotangent),
    Eq (α :=
      Cotangent (R := T ⊗[R] S) (map (F := S →+* T ⊗[R] S) (AlgHom.toRingHom Algebra.TensorProduct.includeRight) I))
      ((tensorCotangentHom R T I :
            (a : T ⊗[R] I.Cotangent) → (map Algebra.TensorProduct.includeRight.toRingHom I).Cotangent)
          a :
        Cotangent (R := T ⊗[R] S) (map (F := S →+* T ⊗[R] S) (AlgHom.toRingHom Algebra.TensorProduct.includeRight) I))
      ((toCotangent (map (F := S →+* T ⊗[R] S) (AlgHom.toRingHom Algebra.TensorProduct.includeRight) I) :
            (a : ↥(map Algebra.TensorProduct.includeRight.toRingHom I)) →
              (map Algebra.TensorProduct.includeRight.toRingHom I).Cotangent)
          (Subtype.mk (p := fun (x : T ⊗[R] S) ↦
            Membership.mem (γ := Ideal (T ⊗[R] S))
              (map (F := S →+* T ⊗[R] S) (AlgHom.toRingHom Algebra.TensorProduct.includeRight) I) x)
            x hx) :
        Cotangent (R := T ⊗[R] S) (map (F := S →+* T ⊗[R] S) (AlgHom.toRingHom Algebra.TensorProduct.includeRight) I))

Tactic:
obtain ⟨y, rfl⟩ := I.map_includeRight_eq.le hx

Premises used: Ideal.map_includeRight_eq, Eq.le, rfl