/-- One step in the proof of the Tietze extension theorem. If `e : C(X, Y)` is a closed embedding
of a topological space into a normal topological space and `f : X →ᵇ ℝ` is a bounded continuous
function, then there exists a bounded continuous function `g : Y →ᵇ ℝ` of the norm `‖g‖ ≤ ‖f‖ / 3`
such that the distance between `g ∘ e` and `f` is at most `(2 / 3) * ‖f‖`. -/
theorem tietze_extension_step (f : X →ᵇ ℝ) (e : C(X, Y)) (he : ClosedEmbedding e) :
    ∃ g : Y →ᵇ ℝ, ‖g‖ ≤ ‖f‖ / 3 ∧ dist (g.compContinuous e) f ≤ 2 / 3 * ‖f‖ := by
  have h3 : (0 : ℝ) < 3 := by norm_num1
  have h23 : 0 < (2 / 3 : ℝ) := by norm_num1
  -- In the trivial case `f = 0`, we take `g = 0`
  rcases eq_or_ne f 0 with (rfl | hf)
  · use 0
    simp
  replace hf : 0 < ‖f‖ := norm_pos_iff.2 hf
  /- Otherwise, the closed sets `e '' (f ⁻¹' (Iic (-‖f‖ / 3)))` and `e '' (f ⁻¹' (Ici (‖f‖ / 3)))`
    are disjoint, hence by Urysohn's lemma there exists a function `g` that is equal to `-‖f‖ / 3`
    on the former set and is equal to `‖f‖ / 3` on the latter set. This function `g` satisfies the
    assertions of the lemma. -/
  have hf3 : -‖f‖ / 3 < ‖f‖ / 3 := (div_lt_div_right h3).2 (Left.neg_lt_self hf)
  have hc₁ : IsClosed (e '' (f ⁻¹' Iic (-‖f‖ / 3))) :=
    he.isClosedMap _ (isClosed_Iic.preimage f.continuous)
  have hc₂ : IsClosed (e '' (f ⁻¹' Ici (‖f‖ / 3))) :=
    he.isClosedMap _ (isClosed_Ici.preimage f.continuous)
  have hd : Disjoint (e '' (f ⁻¹' Iic (-‖f‖ / 3))) (e '' (f ⁻¹' Ici (‖f‖ / 3))) := by
    refine' disjoint_image_of_injective he.inj (Disjoint.preimage _ _)
    rwa [Iic_disjoint_Ici, not_le]
  rcases exists_bounded_mem_Icc_of_closed_of_le hc₁ hc₂ hd hf3.le with ⟨g, hg₁, hg₂, hgf⟩
  refine' ⟨g, _, _⟩
  · refine' (norm_le <| div_nonneg hf.le h3.le).mpr fun y => _
    simpa [abs_le, neg_div] using hgf y
  · refine' (dist_le <| mul_nonneg h23.le hf.le).mpr fun x => _
    have hfx : -‖f‖ ≤ f x ∧ f x ≤ ‖f‖ := by
      simpa only [Real.norm_eq_abs, abs_le] using f.norm_coe_le_norm x
    rcases le_total (f x) (-‖f‖ / 3) with hle₁ | hle₁
    · calc
        |g (e x) - f x| = -‖f‖ / 3 - f x := by
          rw [hg₁ (mem_image_of_mem _ hle₁), Function.const_apply,
            abs_of_nonneg (sub_nonneg.2 hle₁)]
        _ ≤ 2 / 3 * ‖f‖ := by linarith
    · rcases le_total (f x) (‖f‖ / 3) with hle₂ | hle₂
      · simp only [neg_div] at *
        calc
          dist (g (e x)) (f x) ≤ |g (e x)| + |f x| := dist_le_norm_add_norm _ _
          _ ≤ ‖f‖ / 3 + ‖f‖ / 3 := (add_le_add (abs_le.2 <| hgf _) (abs_le.2 ⟨hle₁, hle₂⟩))
          _ = 2 / 3 * ‖f‖ := by linarith
      · 

Proof state:
case inr.intro.intro.intro.refine'_2.inr.inr
X : Type u_1
Y : Type u_2
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : NormalSpace Y
f : X →ᵇ ℝ
e : C(X, Y)
he : ClosedEmbedding ⇑e
h3 : 0 < 3
h23 : 0 < 2 / 3
hf : 0 < ‖f‖
hf3 : -‖f‖ / 3 < ‖f‖ / 3
hc₁ : IsClosed (⇑e '' (⇑f ⁻¹' Iic (-‖f‖ / 3)))
hc₂ : IsClosed (⇑e '' (⇑f ⁻¹' Ici (‖f‖ / 3)))
hd : Disjoint (⇑e '' (⇑f ⁻¹' Iic (-‖f‖ / 3))) (⇑e '' (⇑f ⁻¹' Ici (‖f‖ / 3)))
g : Y →ᵇ ℝ
hg₁ : EqOn (⇑g) (Function.const Y (-‖f‖ / 3)) (⇑e '' (⇑f ⁻¹' Iic (-‖f‖ / 3)))
hg₂ : EqOn (⇑g) (Function.const Y (‖f‖ / 3)) (⇑e '' (⇑f ⁻¹' Ici (‖f‖ / 3)))
hgf : ∀ (x : Y), g x ∈ Icc (-‖f‖ / 3) (‖f‖ / 3)
x : X
hfx : -‖f‖ ≤ f x ∧ f x ≤ ‖f‖
hle₁ : -‖f‖ / 3 ≤ f x
hle₂ : ‖f‖ / 3 ≤ f x
⊢ dist ((compContinuous g e) x) (f x) ≤ 2 / 3 * ‖f‖

Next tactic:
calc
          |g (e x) - f x| = f x - ‖f‖ / 3 := by
            rw [hg₂ (mem_image_of_mem _ hle₂), abs_sub_comm, Function.const_apply,
              abs_of_nonneg (sub_nonneg.2 hle₂)]
          _ ≤ 2 / 3 * ‖f‖ := by linarith