Anti Serre Der Pos

theorem antiSerreDerPos {F : ℍ → ℂ} {k : ℤ} (hMD : MDifferentiable 𝓘(ℂ) 𝓘(ℂ) F)    (hSDF : ResToImagAxis.Pos (serre_D k F))    (hF : ResToImagAxis.EventuallyPos F) : ResToImagAxis.Pos F := by  obtain ⟨_, hSDF_pos⟩ := hSDF  obtain ⟨hF_real, t₀, ht₀_pos, hF_pos⟩ := hF  refine ⟨hF_real, fun t ht => ?_⟩  have key : ∀ s, 0 < s → 0 < (F.resToImagAxis s).re := by    refine  pos_of_deriv_neg_at_zeros ?_ ht₀_pos hF_pos ?_    · intro s hs      exact (continuous_re.continuousAt.comp        (ResToImagAxis.Differentiable F hMD s hs).continuousAt).continuousWithinAt    · intro s hs _ hgs      have hda := hasDerivAt_resToImagAxis_re hMD hs      rw [hda.deriv]      have h_ria : F.resToImagAxis s = F ⟨I * s, by simp [hs]⟩ := by        simp [resToImagAxis, ResToImagAxis, dif_pos hs]      have hz : F (⟨I * s, by simp [hs]⟩ : ℍ) = 0 := by        apply Complex.ext        · rw [zero_re, ← h_ria]; exact hgs        · rw [zero_im, ← h_ria]; exact (hF_real s hs)      have : 0 < ((D F).resToImagAxis s).re := by        simpa [resToImagAxis, ResToImagAxis, dif_pos hs,          serre_D_apply, hz, mul_zero, sub_zero] using hSDF_pos s hs      nlinarith [pi_pos]  exact key t ht