-- 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_1 : S →+* T ⊗[R] S := AlgHom.toRingHom Algebra.TensorProduct.includeRight
x y : TensorProduct.{u_1, u_3, u_2} R T ↥(Submodule.restrictScalars R I)
hy :
  ∃ (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)
            ⋯) :
        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
ha :
  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)
            x)
          ⋯) :
      Cotangent (R := T ⊗[R] S) (map (F := S →+* T ⊗[R] S) (AlgHom.toRingHom Algebra.TensorProduct.includeRight) I))
add :
  ∀ (b : TensorProduct.{u_1, u_3, u_2} R T I.Cotangent)
    (hb :
      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)
            b :
          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)
              ⋯) :
          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)
              ((LinearMap.lTensor T (Submodule.subtype (Submodule.restrictScalars R I)) :
                  (a : T ⊗[R] ↥(Submodule.restrictScalars R I)) → T ⊗[R] S)
                (x + y))
              ⋯) :
          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)
            ((LinearMap.lTensor T (Submodule.subtype (Submodule.restrictScalars R I)) :
                (a : T ⊗[R] ↥(Submodule.restrictScalars R I)) → T ⊗[R] S)
              (x + y))
            ⋯) :
        Cotangent (R := T ⊗[R] S) (map (F := S →+* T ⊗[R] S) (AlgHom.toRingHom Algebra.TensorProduct.includeRight) I))

Tactic:
obtain ⟨b, hb⟩ := hy