-- Lean 4 proof step from Mathlib.Probability.Independence.Conditional (goal)
Goal:
Ω : Type u_1
m' mΩ : MeasurableSpace Ω
inst : StandardBorelSpace Ω
hm' : m' ≤ mΩ
s1 s2 : Set (Set Ω)
hs1 : ∀ s ∈ s1, MeasurableSet s
hs2 : ∀ s ∈ s2, MeasurableSet s
μ : Measure Ω
inst_1 : IsFiniteMeasure μ
hs1_eq :
  ∀ s ∈ s1,
    (ae μ).EventuallyEq (β := ℝ)
      (fun (ω : Ω) ↦
        ENNReal.toReal ((((condExpKernel μ m' : (a : Ω) → Measure Ω) ω : Measure Ω) : (a : Set Ω) → ℝ≥0∞) s))
      μ[s.indicator fun (ω : Ω) ↦ 1 | m']
hs2_eq :
  ∀ s ∈ s2,
    (ae μ).EventuallyEq (β := ℝ)
      (fun (ω : Ω) ↦
        ENNReal.toReal ((((condExpKernel μ m' : (a : Ω) → Measure Ω) ω : Measure Ω) : (a : Set Ω) → ℝ≥0∞) s))
      μ[s.indicator fun (ω : Ω) ↦ 1 | m']
hs12_eq :
  ∀ s ∈ s1,
    ∀ t ∈ s2,
      (ae μ).EventuallyEq (β := ℝ)
        (fun (ω : Ω) ↦
          ENNReal.toReal ((((condExpKernel μ m' : (a : Ω) → Measure Ω) ω : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (s ∩ t)))
        μ[(s ∩ t).indicator fun (ω : Ω) ↦ 1 | m']
s t : Set Ω
hs : s ∈ s1
ht : t ∈ s2
h :
  ∀ᵐ (a : Ω) ∂μ.trim hm',
    (((condExpKernel μ m' : (a : Ω) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (s ∩ t) =
      HMul.hMul (α := ℝ≥0∞) (β := ℝ≥0∞)
        ((((condExpKernel μ m' : (a : Ω) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) s : ℝ≥0∞)
        ((((condExpKernel μ m' : (a : Ω) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) t : ℝ≥0∞)
h'_1 :
  (ae μ).EventuallyEq (β := ℝ≥0∞)
    (fun (x : Ω) ↦ (((condExpKernel μ m' : (a : Ω) → Measure Ω) x : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (s ∩ t))
    fun (x : Ω) ↦
    HMul.hMul (α := ℝ≥0∞) (β := ℝ≥0∞)
      ((((condExpKernel μ m' : (a : Ω) → Measure Ω) x : Measure Ω) : (a : Set Ω) → ℝ≥0∞) s : ℝ≥0∞)
      ((((condExpKernel μ m' : (a : Ω) → Measure Ω) x : Measure Ω) : (a : Set Ω) → ℝ≥0∞) t : ℝ≥0∞)
ω : Ω
hs_eq :
  ENNReal.toReal ((((condExpKernel μ m' : (a : Ω) → Measure Ω) ω : Measure Ω) : (a : Set Ω) → ℝ≥0∞) s) =
    μ[s.indicator fun (ω : Ω) ↦ 1 | m'] ω
ht_eq :
  ENNReal.toReal ((((condExpKernel μ m' : (a : Ω) → Measure Ω) ω : Measure Ω) : (a : Set Ω) → ℝ≥0∞) t) =
    μ[t.indicator fun (ω : Ω) ↦ 1 | m'] ω
hst_eq :
  ENNReal.toReal ((((condExpKernel μ m' : (a : Ω) → Measure Ω) ω : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (s ∩ t)) =
    μ[(s ∩ t).indicator fun (ω : Ω) ↦ 1 | m'] ω
h' :
  (((condExpKernel μ m' : (a : Ω) → Measure Ω) ω : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (s ∩ t) =
    HMul.hMul (α := ℝ≥0∞) (β := ℝ≥0∞)
      ((((condExpKernel μ m' : (a : Ω) → Measure Ω) ω : Measure Ω) : (a : Set Ω) → ℝ≥0∞) s : ℝ≥0∞)
      ((((condExpKernel μ m' : (a : Ω) → Measure Ω) ω : Measure Ω) : (a : Set Ω) → ℝ≥0∞) t : ℝ≥0∞)
h₁ :
  ENNReal.toReal ((((condExpKernel μ m' : (a : Ω) → Measure Ω) ω : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (s ∩ t)) =
    HMul.hMul (α := Ω → ℝ) (β := Ω → ℝ) μ[s.indicator fun (ω : Ω) ↦ 1 | m'] μ[t.indicator fun (ω : Ω) ↦ 1 | m'] ω
⊢ μ[(s ∩ t).indicator fun (ω : Ω) ↦ 1 | m'] ω =
    HMul.hMul (α := Ω → ℝ) (β := Ω → ℝ) μ[s.indicator fun (ω : Ω) ↦ 1 | m'] μ[t.indicator fun (ω : Ω) ↦ 1 | m'] ω
have h₂ : ENNReal.toReal ((((condExpKernel μ m' : (a : Ω) → Measure Ω) ω : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (s ∩ t)) =
  HMul.hMul (α := ℝ) (μ[s.indicator fun (ω : Ω) ↦ 1 | m'] ω) (μ[t.indicator fun (ω : Ω) ↦ 1 | m'] ω)
Tactic: rw [← hst_eq, Pi.mul_apply, ← hs_eq, ← ht_eq, h', ENNReal.toReal_mul]