have h2 := (@TopologicalSpace.IsTopologicalBasis.isOpen_iff ℝ tₗ (f ⁻¹' (Set.Ici (f z)))
   lower_intervals lower_basis).mp inverse_image_open x hx
f : ℝ → ℝ
huc : Continuous f
hlc : Continuous f
z b : ℝ
hz : z ≤ b
x : ℝ
hx : x ∈ f ⁻¹' Set.Ici (f z)
inverse_image_open : IsOpen (f ⁻¹' Set.Ici (f z))
⊢ ∃ y, x < y ∧ Set.Ico x y ⊆ f ⁻¹' Set.Ici (f z)
f : ℝ → ℝ
huc : Continuous f
hlc : Continuous f
z b : ℝ
hz : z ≤ b
x : ℝ
hx : x ∈ f ⁻¹' Set.Ici (f z)
inverse_image_open : IsOpen (f ⁻¹' Set.Ici (f z))
h2 : ∃ t ∈ lower_intervals, x ∈ t ∧ t ⊆ f ⁻¹' Set.Ici (f z)
⊢ ∃ y, x < y ∧ Set.Ico x y ⊆ f ⁻¹' Set.Ici (f z)