-- Module: Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue
Goal:
α : Type u_1
m : MeasurableSpace α
μ ν s : Measure α
f : α → ℝ≥0∞
hf : Measurable f
hadd : μ.singularPart ν + ν.withDensity (μ.rnDeriv ν) = s + ν.withDensity f
this : μ.HaveLebesgueDecomposition ν
hmeas : Measurable (μ.rnDeriv ν)
hadd' : μ = μ.singularPart ν + ν.withDensity (μ.rnDeriv ν)
S : Set α
hS₁ : MeasurableSet S
hS₂ : Eq (α := ℝ≥0∞) ((s : (a : Set α) → ℝ≥0∞) S : ℝ≥0∞) (0 : ℝ≥0∞)
hS₃ : Eq (α := ℝ≥0∞) ((ν : (a : Set α) → ℝ≥0∞) Sᶜ : ℝ≥0∞) (0 : ℝ≥0∞)
T : Set α
hT₁ : MeasurableSet T
hT₂ : Eq (α := ℝ≥0∞) ((μ.singularPart ν : (a : Set α) → ℝ≥0∞) T : ℝ≥0∞) (0 : ℝ≥0∞)
hT₃ : Eq (α := ℝ≥0∞) ((ν : (a : Set α) → ℝ≥0∞) Tᶜ : ℝ≥0∞) (0 : ℝ≥0∞)
hνinter : Eq (α := ℝ≥0∞) ((ν : (a : Set α) → ℝ≥0∞) (S ∩ T)ᶜ : ℝ≥0∞) (0 : ℝ≥0∞)
A : Set α
hA : MeasurableSet A
hs : Eq (α := ℝ≥0∞) ((s : (a : Set α) → ℝ≥0∞) (A ∩ (S ∩ T)) : ℝ≥0∞) (0 : ℝ≥0∞)
hsing : Eq (α := ℝ≥0∞) ((μ.singularPart ν : (a : Set α) → ℝ≥0∞) (A ∩ (S ∩ T)) : ℝ≥0∞) (0 : ℝ≥0∞)
h :
  Eq (α := ℝ≥0∞) ((ν.withDensity f : (a : Set α) → ℝ≥0∞) (A ∩ (S ∩ T)) : ℝ≥0∞)
    (((ν.withDensity (μ.rnDeriv ν)).restrict (S ∩ T) : (a : Set α) → ℝ≥0∞) A : ℝ≥0∞)
h₁ :
  Eq (α := ℝ≥0∞) ((ν.withDensity f : (a : Set α) → ℝ≥0∞) (A ∩ (S ∩ T)) : ℝ≥0∞)
    ((ν.withDensity (μ.rnDeriv ν) : (a : Set α) → ℝ≥0∞) (A ∩ (S ∩ T)) : ℝ≥0∞)
h₂ :
  HAdd.hAdd (α := ℝ≥0∞) (β := ℝ≥0∞) ((ν.withDensity f : (a : Set α) → ℝ≥0∞) (A ∩ (S ∩ T)) : ℝ≥0∞) (0 : ℝ≥0∞) =
    (ν.withDensity (μ.rnDeriv ν) : (a : Set α) → ℝ≥0∞) (A ∩ (S ∩ T))
h₃ :
  HAdd.hAdd (α := ℝ≥0∞) (β := ℝ≥0∞) ((ν.withDensity f : (a : Set α) → ℝ≥0∞) (A ∩ (S ∩ T)) : ℝ≥0∞)
      ((s : (a : Set α) → ℝ≥0∞) (A ∩ (S ∩ T)) : ℝ≥0∞) =
    (ν.withDensity (μ.rnDeriv ν) : (a : Set α) → ℝ≥0∞) (A ∩ (S ∩ T))
h₄ :
  Eq (α := ℝ≥0∞) ((ν.withDensity f + s : (a : Set α) → ℝ≥0∞) (A ∩ (S ∩ T)) : ℝ≥0∞)
    ((ν.withDensity (μ.rnDeriv ν) : (a : Set α) → ℝ≥0∞) (A ∩ (S ∩ T)) : ℝ≥0∞)
h₅ :
  Eq (α := ℝ≥0∞) ((s + ν.withDensity f : (a : Set α) → ℝ≥0∞) (A ∩ (S ∩ T)) : ℝ≥0∞)
    ((ν.withDensity (μ.rnDeriv ν) : (a : Set α) → ℝ≥0∞) (A ∩ (S ∩ T)) : ℝ≥0∞)
h₆ :
  Eq (α := ℝ≥0∞) ((μ.singularPart ν + ν.withDensity (μ.rnDeriv ν) : (a : Set α) → ℝ≥0∞) (A ∩ (S ∩ T)) : ℝ≥0∞)
    ((ν.withDensity (μ.rnDeriv ν) : (a : Set α) → ℝ≥0∞) (A ∩ (S ∩ T)) : ℝ≥0∞)
h₇ :
  HAdd.hAdd (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ.singularPart ν : (a : Set α) → ℝ≥0∞) (A ∩ (S ∩ T)) : ℝ≥0∞)
      ((ν.withDensity (μ.rnDeriv ν) : (a : Set α) → ℝ≥0∞) (A ∩ (S ∩ T)) : ℝ≥0∞) =
    (ν.withDensity (μ.rnDeriv ν) : (a : Set α) → ℝ≥0∞) (A ∩ (S ∩ T))
h₈ :
  HAdd.hAdd (α := ℝ≥0∞) (β := ℝ≥0∞) (0 : ℝ≥0∞)
      ((ν.withDensity (μ.rnDeriv ν) : (a : Set α) → ℝ≥0∞) (A ∩ (S ∩ T)) : ℝ≥0∞) =
    (ν.withDensity (μ.rnDeriv ν) : (a : Set α) → ℝ≥0∞) (A ∩ (S ∩ T))
h₉ h₁₀ :
  Eq (α := ℝ≥0∞) ((ν.withDensity (μ.rnDeriv ν) : (a : Set α) → ℝ≥0∞) (A ∩ (S ∩ T)) : ℝ≥0∞)
    ((ν.withDensity (μ.rnDeriv ν) : (a : Set α) → ℝ≥0∞) (A ∩ (S ∩ T)) : ℝ≥0∞)
⊢ Eq (α := ℝ≥0∞) (((ν.withDensity f).restrict (S ∩ T) : (a : Set α) → ℝ≥0∞) A : ℝ≥0∞)
    (((ν.withDensity (μ.rnDeriv ν)).restrict (S ∩ T) : (a : Set α) → ℝ≥0∞) A : ℝ≥0∞)

Tactic:
rw [restrict_apply hA, restrict_apply hA, ← add_zero (ν.withDensity f (A ∩ (S ∩ T))), ← hs, ←
      add_apply, add_comm, ← hadd, add_apply, hsing, zero_add]

Premises used: MeasureTheory.Measure.withDensity, MeasureTheory.Measure.restrict_apply, MeasureTheory.Measure.add_apply, zero_add, add_comm, Inter.inter, add_zero, DFunLike.coe