-- Module: Mathlib.RingTheory.Kaehler.Basic
Goal:
R : Type u
S : Type v
inst : CommRing R
inst_1 : CommRing S
inst_2 : Algebra R S
x : S ⊗[R] S
hx_1 : Membership.mem (γ := Ideal (S ⊗[R] S)) (ideal R S) x
hx :
  Membership.mem (γ := Submodule S (S ⊗[R] S))
    (Submodule.span S (Set.range fun (s : S) ↦ HSub.hSub (α := S ⊗[R] S) (1 ⊗ₜ[R] s) (s ⊗ₜ[R] 1))) x
refine_1 :
  ∀ (x : S ⊗[R] S)
    (h :
      Membership.mem (γ := Set (S ⊗[R] S)) (Set.range fun (s : S) ↦ HSub.hSub (α := S ⊗[R] S) (1 ⊗ₜ[R] s) (s ⊗ₜ[R] 1))
        x),
    ∃ (hx : Membership.mem (γ := Ideal (S ⊗[R] S)) (ideal R S) x),
      Membership.mem (α := Ω[S⁄R]) (γ := Submodule S Ω[S⁄R])
        (Submodule.span S (Set.range (ι := S) (⇑(D R S) : (a : S) → Ω[S⁄R])))
        ((fromIdeal R S : (a : ↥(ideal R S)) → Ω[S⁄R])
            (Subtype.mk (p := fun (x : S ⊗[R] S) ↦ Membership.mem (γ := Ideal (S ⊗[R] S)) (ideal R S) x) x hx) :
          Ω[S⁄R])
refine_2 :
  ∃ (hx : Membership.mem (γ := Ideal (S ⊗[R] S)) (ideal R S) 0),
    Membership.mem (α := Ω[S⁄R]) (γ := Submodule S Ω[S⁄R])
      (Submodule.span S (Set.range (ι := S) (⇑(D R S) : (a : S) → Ω[S⁄R])))
      ((fromIdeal R S : (a : ↥(ideal R S)) → Ω[S⁄R])
          (Subtype.mk (p := fun (x : S ⊗[R] S) ↦ Membership.mem (γ := Ideal (S ⊗[R] S)) (ideal R S) x) 0 hx) :
        Ω[S⁄R])
refine_3 :
  ∀ (x y : S ⊗[R] S)
    (hx :
      Membership.mem (γ := Submodule S (S ⊗[R] S))
        (Submodule.span S (Set.range fun (s : S) ↦ HSub.hSub (α := S ⊗[R] S) (1 ⊗ₜ[R] s) (s ⊗ₜ[R] 1))) x)
    (hy :
      Membership.mem (γ := Submodule S (S ⊗[R] S))
        (Submodule.span S (Set.range fun (s : S) ↦ HSub.hSub (α := S ⊗[R] S) (1 ⊗ₜ[R] s) (s ⊗ₜ[R] 1))) y),
    (∃ (hx : Membership.mem (γ := Ideal (S ⊗[R] S)) (ideal R S) x),
        Membership.mem (α := Ω[S⁄R]) (γ := Submodule S Ω[S⁄R])
          (Submodule.span S (Set.range (ι := S) (⇑(D R S) : (a : S) → Ω[S⁄R])))
          ((fromIdeal R S : (a : ↥(ideal R S)) → Ω[S⁄R])
              (Subtype.mk (p := fun (x : S ⊗[R] S) ↦ Membership.mem (γ := Ideal (S ⊗[R] S)) (ideal R S) x) x hx) :
            Ω[S⁄R])) →
      (∃ (hx : Membership.mem (γ := Ideal (S ⊗[R] S)) (ideal R S) y),
          Membership.mem (α := Ω[S⁄R]) (γ := Submodule S Ω[S⁄R])
            (Submodule.span S (Set.range (ι := S) (⇑(D R S) : (a : S) → Ω[S⁄R])))
            ((fromIdeal R S : (a : ↥(ideal R S)) → Ω[S⁄R])
                (Subtype.mk (p := fun (x : S ⊗[R] S) ↦ Membership.mem (γ := Ideal (S ⊗[R] S)) (ideal R S) x) y hx) :
              Ω[S⁄R])) →
        ∃ (hx : Membership.mem (γ := Ideal (S ⊗[R] S)) (ideal R S) (x + y)),
          Membership.mem (α := Ω[S⁄R]) (γ := Submodule S Ω[S⁄R])
            (Submodule.span S (Set.range (ι := S) (⇑(D R S) : (a : S) → Ω[S⁄R])))
            ((fromIdeal R S : (a : ↥(ideal R S)) → Ω[S⁄R])
                (Subtype.mk (p := fun (x : S ⊗[R] S) ↦ Membership.mem (γ := Ideal (S ⊗[R] S)) (ideal R S) x) (x + y)
                  hx) :
              Ω[S⁄R])
refine_4 :
  ∀ (a : S) (x : S ⊗[R] S)
    (hx :
      Membership.mem (γ := Submodule S (S ⊗[R] S))
        (Submodule.span S (Set.range fun (s : S) ↦ HSub.hSub (α := S ⊗[R] S) (1 ⊗ₜ[R] s) (s ⊗ₜ[R] 1))) x),
    (∃ (hx : Membership.mem (γ := Ideal (S ⊗[R] S)) (ideal R S) x),
        Membership.mem (α := Ω[S⁄R]) (γ := Submodule S Ω[S⁄R])
          (Submodule.span S (Set.range (ι := S) (⇑(D R S) : (a : S) → Ω[S⁄R])))
          ((fromIdeal R S : (a : ↥(ideal R S)) → Ω[S⁄R])
              (Subtype.mk (p := fun (x : S ⊗[R] S) ↦ Membership.mem (γ := Ideal (S ⊗[R] S)) (ideal R S) x) x hx) :
            Ω[S⁄R])) →
      ∃ (hx : Membership.mem (γ := Ideal (S ⊗[R] S)) (ideal R S) (a • x)),
        Membership.mem (α := Ω[S⁄R]) (γ := Submodule S Ω[S⁄R])
          (Submodule.span S (Set.range (ι := S) (⇑(D R S) : (a : S) → Ω[S⁄R])))
          ((fromIdeal R S : (a : ↥(ideal R S)) → Ω[S⁄R])
              (Subtype.mk (p := fun (x : S ⊗[R] S) ↦ Membership.mem (γ := Ideal (S ⊗[R] S)) (ideal R S) x) (a • x) hx) :
            Ω[S⁄R])
⊢ ∃ (hx : Membership.mem (γ := Ideal (S ⊗[R] S)) (ideal R S) x),
    Membership.mem (α := Ω[S⁄R]) (γ := Submodule S Ω[S⁄R])
      (Submodule.span S (Set.range (ι := S) (⇑(D R S) : (a : S) → Ω[S⁄R])))
      ((fromIdeal R S : (a : ↥(ideal R S)) → Ω[S⁄R])
          (Subtype.mk (p := fun (x : S ⊗[R] S) ↦ Membership.mem (γ := Ideal (S ⊗[R] S)) (ideal R S) x) x hx) :
        Ω[S⁄R])

Tactic:
refine Submodule.span_induction ?_ ?_ ?_ ?_ hx