/-- **Divergence theorem** for functions on the plane along rectangles. It is formulated in terms of
two functions `f g : ℝ × ℝ → E` and an integral over `Icc a b = [a.1, b.1] × [a.2, b.2]`, where
`a b : ℝ × ℝ`, `a ≤ b`. When thinking of `f` and `g` as the two coordinates of a single function
`F : ℝ × ℝ → E × E` and when `E = ℝ`, this is the usual statement that the integral of the
divergence of `F` inside the rectangle equals the integral of the normal derivative of `F` along the
boundary.

See also `MeasureTheory.integral2_divergence_prod_of_hasFDerivWithinAt_off_countable` for a
version that does not assume `a ≤ b` and uses iterated interval integral instead of the integral
over `Icc a b`. -/
theorem integral_divergence_prod_Icc_of_hasFDerivWithinAt_off_countable_of_le (f g : ℝ × ℝ → E)
    (f' g' : ℝ × ℝ → ℝ × ℝ →L[ℝ] E) (a b : ℝ × ℝ) (hle : a ≤ b) (s : Set (ℝ × ℝ)) (hs : s.Countable)
    (Hcf : ContinuousOn f (Icc a b)) (Hcg : ContinuousOn g (Icc a b))
    (Hdf : ∀ x ∈ Ioo a.1 b.1 ×ˢ Ioo a.2 b.2 \ s, HasFDerivAt f (f' x) x)
    (Hdg : ∀ x ∈ Ioo a.1 b.1 ×ˢ Ioo a.2 b.2 \ s, HasFDerivAt g (g' x) x)
    (Hi : IntegrableOn (fun x => f' x (1, 0) + g' x (0, 1)) (Icc a b)) :
    (∫ x in Icc a b, f' x (1, 0) + g' x (0, 1)) =
      (((∫ x in a.1..b.1, g (x, b.2)) - ∫ x in a.1..b.1, g (x, a.2)) +
          ∫ y in a.2..b.2, f (b.1, y)) -
        ∫ y in a.2..b.2, f (a.1, y) :=
  let e : (ℝ × ℝ) ≃L[ℝ] ℝ² := (ContinuousLinearEquiv.finTwoArrow ℝ ℝ).symm
  calc
    (∫ x in Icc a b, f' x (1, 0) + g' x (0, 1)) =
        ∑ i : Fin 2,
          ((∫ x in Icc (e a ∘ i.succAbove) (e b ∘ i.succAbove),
              ![f, g] i (e.symm <| i.insertNth (e b i) x)) -
            ∫ x in Icc (e a ∘ i.succAbove) (e b ∘ i.succAbove),
              ![f, g] i (e.symm <| i.insertNth (e a i) x)) := by
      refine' integral_divergence_of_hasFDerivWithinAt_off_countable_of_equiv e _ _ ![f, g]
        ![f', g'] s hs a b hle _ (fun x hx => _) _ _ Hi
      · exact fun x y => (OrderIso.finTwoArrowIso ℝ).symm.le_iff_le
      · exact (volume_preserving_finTwoArrow ℝ).symm _
      · exact Fin.forall_fin_two.2 ⟨Hcf, Hcg⟩
      · rw [Icc_prod_eq, interior_prod_eq, interior_Icc, interior_Icc] at hx
        exact Fin.forall_fin_two.2 ⟨Hdf x hx, Hdg x hx⟩
      · intro x; rw [Fin.sum_univ_two]; rfl
    _ = ((∫ y in Icc a.2 b.2, f (b.1, y)) - ∫ y in Icc a.2 b.2, f (a.1, y)) +
          ((∫ x in Icc a.1 b.1, g (x, b.2)) - ∫ x in Icc a.1 b.1, g (x, a.2)) := by
      have : ∀ (a b : ℝ¹) (f : ℝ¹ → E),
          ∫ x in Icc a b, f x = ∫ x in Icc (a 0) (b 0), f fun _ => x := fun a b f ↦ by
        convert (((volume_preserving_funUnique (Fin 1) ℝ).symm _).set_integral_preimage_emb
          (MeasurableEquiv.measurableEmbedding _) f _).symm
        exact ((OrderIso.funUnique (Fin 1) ℝ).symm.preimage_Icc a b).symm
      simp only [Fin.sum_univ_two, this]
      

Proof state:
E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f g : ℝ × ℝ → E
f' g' : ℝ × ℝ → ℝ × ℝ →L[ℝ] E
a b : ℝ × ℝ
hle : a ≤ b
s : Set (ℝ × ℝ)
hs : Set.Countable s
Hcf : ContinuousOn f (Set.Icc a b)
Hcg : ContinuousOn g (Set.Icc a b)
Hdf : ∀ x ∈ Set.Ioo a.1 b.1 ×ˢ Set.Ioo a.2 b.2 \ s, HasFDerivAt f (f' x) x
Hdg : ∀ x ∈ Set.Ioo a.1 b.1 ×ˢ Set.Ioo a.2 b.2 \ s, HasFDerivAt g (g' x) x
Hi : IntegrableOn (fun x => (f' x) (1, 0) + (g' x) (0, 1)) (Set.Icc a b)
e : (ℝ × ℝ) ≃L[ℝ] Fin 2 → ℝ := ContinuousLinearEquiv.symm (ContinuousLinearEquiv.finTwoArrow ℝ ℝ)
this :
  ∀ (a b : Fin 1 → ℝ) (f : (Fin 1 → ℝ) → E),
    ∫ (x : Fin 1 → ℝ) in Set.Icc a b, f x = ∫ (x : ℝ) in Set.Icc (a 0) (b 0), f fun x_1 => x
⊢ ((∫ (x : ℝ) in
          Set.Icc (((ContinuousLinearEquiv.symm (ContinuousLinearEquiv.finTwoArrow ℝ ℝ)) a ∘ Fin.succAbove 0) 0)
            (((ContinuousLinearEquiv.symm (ContinuousLinearEquiv.finTwoArrow ℝ ℝ)) b ∘ Fin.succAbove 0) 0),
          Matrix.vecCons f ![g] 0
            ((ContinuousLinearEquiv.symm (ContinuousLinearEquiv.symm (ContinuousLinearEquiv.finTwoArrow ℝ ℝ)))
              (Fin.insertNth 0 ((ContinuousLinearEquiv.symm (ContinuousLinearEquiv.finTwoArrow ℝ ℝ)) b 0) fun x_1 =>
                x))) -
        ∫ (x : ℝ) in
          Set.Icc (((ContinuousLinearEquiv.symm (ContinuousLinearEquiv.finTwoArrow ℝ ℝ)) a ∘ Fin.succAbove 0) 0)
            (((ContinuousLinearEquiv.symm (ContinuousLinearEquiv.finTwoArrow ℝ ℝ)) b ∘ Fin.succAbove 0) 0),
          Matrix.vecCons f ![g] 0
            ((ContinuousLinearEquiv.symm (ContinuousLinearEquiv.symm (ContinuousLinearEquiv.finTwoArrow ℝ ℝ)))
              (Fin.insertNth 0 ((ContinuousLinearEquiv.symm (ContinuousLinearEquiv.finTwoArrow ℝ ℝ)) a 0) fun x_1 =>
                x))) +
      ((∫ (x : ℝ) in
          Set.Icc (((ContinuousLinearEquiv.symm (ContinuousLinearEquiv.finTwoArrow ℝ ℝ)) a ∘ Fin.succAbove 1) 0)
            (((ContinuousLinearEquiv.symm (ContinuousLinearEquiv.finTwoArrow ℝ ℝ)) b ∘ Fin.succAbove 1) 0),
          Matrix.vecCons f ![g] 1
            ((ContinuousLinearEquiv.symm (ContinuousLinearEquiv.symm (ContinuousLinearEquiv.finTwoArrow ℝ ℝ)))
              (Fin.insertNth 1 ((ContinuousLinearEquiv.symm (ContinuousLinearEquiv.finTwoArrow ℝ ℝ)) b 1) fun x_1 =>
                x))) -
        ∫ (x : ℝ) in
          Set.Icc (((ContinuousLinearEquiv.symm (ContinuousLinearEquiv.finTwoArrow ℝ ℝ)) a ∘ Fin.succAbove 1) 0)
            (((ContinuousLinearEquiv.symm (ContinuousLinearEquiv.finTwoArrow ℝ ℝ)) b ∘ Fin.succAbove 1) 0),
          Matrix.vecCons f ![g] 1
            ((ContinuousLinearEquiv.symm (ContinuousLinearEquiv.symm (ContinuousLinearEquiv.finTwoArrow ℝ ℝ)))
              (Fin.insertNth 1 ((ContinuousLinearEquiv.symm (ContinuousLinearEquiv.finTwoArrow ℝ ℝ)) a 1) fun x_1 =>
                x))) =
    ((∫ (y : ℝ) in Set.Icc a.2 b.2, f (b.1, y)) - ∫ (y : ℝ) in Set.Icc a.2 b.2, f (a.1, y)) +
      ((∫ (x : ℝ) in Set.Icc a.1 b.1, g (x, b.2)) - ∫ (x : ℝ) in Set.Icc a.1 b.1, g (x, a.2))

Next tactic:
rfl