-- Module: Mathlib.RingTheory.Flat.FaithfullyFlat.Basic
Goal:
R : Type u
M : Type v
inst : CommRing R
inst_1 : AddCommGroup M
inst_2 : Module R M
N1 : Type u_1
inst_3 : AddCommGroup N1
inst_4 : Module R N1
N2 : Type u_2
inst_5 : AddCommGroup N2
inst_6 : Module R N2
N3 : Type u_3
inst_7 : AddCommGroup N3
inst_8 : Module R N3
l12 : N1 →ₗ[R] N2
l23 : N2 →ₗ[R] N3
fl : FaithfullyFlat R M
ex :
  Function.Exact (M := N1 ⊗[R] M) (N := N2 ⊗[R] M) (P := N3 ⊗[R] M)
    (⇑(LinearMap.rTensor M l12) : (a : N1 ⊗[R] M) → N2 ⊗[R] M)
    (⇑(LinearMap.rTensor M l23) : (a : N2 ⊗[R] M) → N3 ⊗[R] M)
complex : l12.range ≤ l23.ker
H : Type u_2 := ↥l23.ker ⧸ (Submodule.inclusion complex).range
e : H ⊗[R] M ≃ₗ[R]
  HasQuotient.Quotient (TensorProduct.{u, u_2, v} R (↥l23.ker) M) (B :=
    Submodule R (TensorProduct.{u, u_2, v} R (↥l23.ker) M))
    (LinearMap.range (TensorProduct.map (Submodule.inclusion complex).range.subtype LinearMap.id)) :=
  quotientTensorEquiv M (Submodule.inclusion complex).range
x : TensorProduct.{u, u_2, v} R (↥l23.ker) M
zero :
  ∃ (a : TensorProduct.{u, u_2, v} R (↥(Submodule.inclusion complex).range) M),
    Eq (α := TensorProduct.{u, u_2, v} R (↥l23.ker) M)
      ((TensorProduct.map (Submodule.inclusion complex).range.subtype LinearMap.id :
            (a : ↥(Submodule.inclusion complex).range ⊗[R] M) → ↥l23.ker ⊗[R] M)
          a :
        TensorProduct.{u, u_2, v} R (↥l23.ker) M)
      (0 : TensorProduct.{u, u_2, v} R (↥l23.ker) M)
tmul :
  ∀ (x : ↥l23.ker) (m : M),
    ∃ (a : TensorProduct.{u, u_2, v} R (↥(Submodule.inclusion complex).range) M),
      Eq (α := TensorProduct.{u, u_2, v} R (↥l23.ker) M)
        ((TensorProduct.map (Submodule.inclusion complex).range.subtype LinearMap.id :
              (a : ↥(Submodule.inclusion complex).range ⊗[R] M) → ↥l23.ker ⊗[R] M)
            a :
          TensorProduct.{u, u_2, v} R (↥l23.ker) M)
        (x ⊗ₜ[R] m : TensorProduct.{u, u_2, v} R (↥l23.ker) M)
add :
  ∀ (x y : TensorProduct.{u, u_2, v} R (↥l23.ker) M)
    (hx :
      ∃ (a : TensorProduct.{u, u_2, v} R (↥(Submodule.inclusion complex).range) M),
        (TensorProduct.map (Submodule.inclusion complex).range.subtype LinearMap.id :
              (a : ↥(Submodule.inclusion complex).range ⊗[R] M) → ↥l23.ker ⊗[R] M)
            a =
          x)
    (hy :
      ∃ (a : TensorProduct.{u, u_2, v} R (↥(Submodule.inclusion complex).range) M),
        (TensorProduct.map (Submodule.inclusion complex).range.subtype LinearMap.id :
              (a : ↥(Submodule.inclusion complex).range ⊗[R] M) → ↥l23.ker ⊗[R] M)
            a =
          y),
    ∃ (a : TensorProduct.{u, u_2, v} R (↥(Submodule.inclusion complex).range) M),
      (TensorProduct.map (Submodule.inclusion complex).range.subtype LinearMap.id :
            (a : ↥(Submodule.inclusion complex).range ⊗[R] M) → ↥l23.ker ⊗[R] M)
          a =
        x + y
⊢ ∃ (a : TensorProduct.{u, u_2, v} R (↥(Submodule.inclusion complex).range) M),
    (TensorProduct.map (Submodule.inclusion complex).range.subtype LinearMap.id :
          (a : ↥(Submodule.inclusion complex).range ⊗[R] M) → ↥l23.ker ⊗[R] M)
        a =
      x

Tactic:
induction x using TensorProduct.induction_on with
  | zero => exact ⟨0, by simp⟩
  -- let `x ⊗ m` be an element in `ker l23 ⊗ M`, then `x ⊗ m` is in the kernel of `l23 ⊗ 𝟙M`.
  -- Since `N1 ⊗ M -l12 ⊗ M-> N2 ⊗ M -l23 ⊗ M-> N3 ⊗ M` is exact, we have that `x ⊗ m` is in
  -- the range of `l12 ⊗ 𝟙M`, i.e. `x ⊗ m = (l12 ⊗ 𝟙M) y` for some `y ∈ N1 ⊗ M` as elements of
  -- `N2 ⊗ M`. We need to prove that `x ⊗ m = (l12 ⊗ 𝟙M) y` still holds in `(ker l23) ⊗ M`.
  -- This is okay because `M` is flat and `ker l23 -> N2` is injective.
  | tmul x m =>
    rcases x with ⟨x, (hx : l23 x = 0)⟩
    have mem : x ⊗ₜ[R] m ∈ LinearMap.ker (l23.rTensor M) := by simp [hx]
    rw [LinearMap.exact_iff.1 ex] at mem
    obtain ⟨y, hy⟩ := mem
    refine ⟨LinearMap.rTensor M (LinearMap.rangeRestrict _ ∘ₗ LinearMap.rangeRestrict l12) y,
      Module.Flat.rTensor_preserves_injective_linearMap (LinearMap.ker l23).subtype
      Subtype.val_injective ?_⟩
    simp only [LinearMap.comp_codRestrict, LinearMap.rTensor_tmul, Submodule.coe_subtype, ← hy]
    rw [← LinearMap.comp_apply, ← LinearMap.rTensor_def, ← LinearMap.rTensor_comp,
      ← LinearMap.comp_apply, ← LinearMap.rTensor_comp, LinearMap.comp_assoc,
      LinearMap.subtype_comp_codRestrict, ← LinearMap.comp_assoc, Submodule.subtype_comp_inclusion,
      LinearMap.subtype_comp_codRestrict]
  | add x y hx hy =>
    obtain ⟨x, rfl⟩ := hx; obtain ⟨y, rfl⟩ := hy
    exact ⟨x + y, by simp⟩

Premises used: Submodule.subtype_comp_inclusion, LinearMap.subtype_comp_codRestrict, LinearMap.rTensor_tmul, LinearMap.comp_assoc, LinearMap.comp_apply, LinearMap.comp_codRestrict, LinearMap.rTensor_comp, LinearMap.rTensor_def, Iff.mp, Submodule.coe_subtype, TensorProduct.tmul, LinearMap.exact_iff, rfl