/-- Suppose that a function `f : ℂ → E` is continuous on a closed rectangle with opposite corners at
`z w : ℂ`, is *real* differentiable at all but countably many points of the corresponding open
rectangle, and $\frac{\partial f}{\partial \bar z}$ is integrable on this rectangle. Then the
integral of `f` over the boundary of the rectangle is equal to the integral of
$2i\frac{\partial f}{\partial \bar z}=i\frac{\partial f}{\partial x}-\frac{\partial f}{\partial y}$
over the rectangle. -/
theorem integral_boundary_rect_of_hasFDerivAt_real_off_countable (f : ℂ → E) (f' : ℂ → ℂ →L[ℝ] E)
    (z w : ℂ) (s : Set ℂ) (hs : s.Countable)
    (Hc : ContinuousOn f ([[z.re, w.re]] ×ℂ [[z.im, w.im]]))
    (Hd : ∀ x ∈ Ioo (min z.re w.re) (max z.re w.re) ×ℂ Ioo (min z.im w.im) (max z.im w.im) \ s,
      HasFDerivAt f (f' x) x)
    (Hi : IntegrableOn (fun z => I • f' z 1 - f' z I) ([[z.re, w.re]] ×ℂ [[z.im, w.im]])) :
    (∫ x : ℝ in z.re..w.re, f (x + z.im * I)) - (∫ x : ℝ in z.re..w.re, f (x + w.im * I)) +
      I • (∫ y : ℝ in z.im..w.im, f (re w + y * I)) -
      I • ∫ y : ℝ in z.im..w.im, f (re z + y * I) =
      ∫ x : ℝ in z.re..w.re, ∫ y : ℝ in z.im..w.im, I • f' (x + y * I) 1 - f' (x + y * I) I := by
  set e : (ℝ × ℝ) ≃L[ℝ] ℂ := equivRealProdClm.symm
  have he : ∀ x y : ℝ, ↑x + ↑y * I = e (x, y) := fun x y => (mk_eq_add_mul_I x y).symm
  have he₁ : e (1, 0) = 1 := rfl; have he₂ : e (0, 1) = I := rfl
  simp only [he] at *
  set F : ℝ × ℝ → E := f ∘ e
  set F' : ℝ × ℝ → ℝ × ℝ →L[ℝ] E := fun p => (f' (e p)).comp (e : ℝ × ℝ →L[ℝ] ℂ)
  have hF' : ∀ p : ℝ × ℝ, (-(I • F' p)) (1, 0) + F' p (0, 1) = -(I • f' (e p) 1 - f' (e p) I) := by
    rintro ⟨x, y⟩
    simp only [ContinuousLinearMap.neg_apply, ContinuousLinearMap.smul_apply,
      ContinuousLinearMap.comp_apply, ContinuousLinearEquiv.coe_coe, he₁, he₂, neg_add_eq_sub,
      neg_sub]
  

Proof state:
E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ → E
f' : ℂ → ℂ →L[ℝ] E
z w : ℂ
s : Set ℂ
hs : Set.Countable s
Hc : ContinuousOn f ([[z.re, w.re]] ×ℂ [[z.im, w.im]])
Hd : ∀ x ∈ Ioo (min z.re w.re) (max z.re w.re) ×ℂ Ioo (min z.im w.im) (max z.im w.im) \ s, HasFDerivAt f (f' x) x
Hi : IntegrableOn (fun z => I • (f' z) 1 - (f' z) I) ([[z.re, w.re]] ×ℂ [[z.im, w.im]])
e : (ℝ × ℝ) ≃L[ℝ] ℂ := ContinuousLinearEquiv.symm equivRealProdClm
he₁ : (ContinuousLinearEquiv.symm equivRealProdClm) (1, 0) = 1
he₂ : (ContinuousLinearEquiv.symm equivRealProdClm) (0, 1) = I
he : ℝ → ℝ → True
F : ℝ × ℝ → E := f ∘ ⇑e
F' : ℝ × ℝ → ℝ × ℝ →L[ℝ] E := fun p => ContinuousLinearMap.comp (f' (e p)) ↑e
hF' : ∀ (p : ℝ × ℝ), (-(I • F' p)) (1, 0) + (F' p) (0, 1) = -(I • (f' (e p)) 1 - (f' (e p)) I)
⊢ (((∫ (x : ℝ) in z.re..w.re, f ((ContinuousLinearEquiv.symm equivRealProdClm) (x, z.im))) -
          ∫ (x : ℝ) in z.re..w.re, f ((ContinuousLinearEquiv.symm equivRealProdClm) (x, w.im))) +
        I • ∫ (y : ℝ) in z.im..w.im, f ((ContinuousLinearEquiv.symm equivRealProdClm) (w.re, y))) -
      I • ∫ (y : ℝ) in z.im..w.im, f ((ContinuousLinearEquiv.symm equivRealProdClm) (z.re, y)) =
    ∫ (x : ℝ) in z.re..w.re,
      ∫ (y : ℝ) in z.im..w.im,
        I • (f' ((ContinuousLinearEquiv.symm equivRealProdClm) (x, y))) 1 -
          (f' ((ContinuousLinearEquiv.symm equivRealProdClm) (x, y))) I

Next tactic:
set R : Set (ℝ × ℝ) := [[z.re, w.re]] ×ˢ [[w.im, z.im]]