have hfut {u : ℕ} (hxu : x ≤ u) (hut : u < t) (hfu : trunc n u = u)
  (hux' : u ≤ round (y - round (y - x)))
: trunc n (t - u) = t - u := by
  have hfuxy : trunc n (y - round (y - x) - u) = y - round (y - x) - u := by
    have hxyx : x < round (y - round (y - x)) := calc
      x ≤ u                         := hxu
      _ < t                         := hut
      _ ≤ y - round (y - x)         := htx'
      _ = round (y - round (y - x)) := hfx'.symm
    rewrite [← hfx']
    exact s₃ npos hfaithful₀ hfaithful₁ hfy hfx hfu hxy hxu hyxulp hxyx hux'
  rewrite [ht] at hut htx' ⊢
  replace hfx' := trunc_eq_of_round_eq npos hfaithful₁ hfx'
  exact s₁ npos hut.le htx' hfu hfx' (trunc_round npos hfaithful₁ _) hfuxy
round : ℕ → ℕ
x y z w s t n : ℕ
npos : 0 < n
hfaithful₀ : faithful₀ n round
hfaithful₁ : faithful₁ n round
hml : sub_left_monotonic n round
hfx : trunc n x = x
hfy : trunc n y = y
hfz : trunc n z = z
hfw : trunc n w = w
hzx : z ≤ x
hzy : z ≤ y
hyxulp : ulp n y ≤ 2 * x
ht : t = round (y - round (y - z))
hs₁ : x < t → s = round (x + round (t - x))
hs₂ : t ≤ x → s = round (x - round (x - t))
hxw : x ≤ w
hwy : w ≤ y
hxy : x ≤ y
ht' : t = y - round (y - z)
htx' : t ≤ y - round (y - x)
hzy' : round (y - z) ≤ y
hft : trunc n t = t
hfx' : round (y - round (y - x)) = y - round (y - x)
⊢ trunc n (w - s) = w - s ∧ trunc n (s - w) = s - w
round : ℕ → ℕ
x y z w s t n : ℕ
npos : 0 < n
hfaithful₀ : faithful₀ n round
hfaithful₁ : faithful₁ n round
hml : sub_left_monotonic n round
hfx : trunc n x = x
hfy : trunc n y = y
hfz : trunc n z = z
hfw : trunc n w = w
hzx : z ≤ x
hzy : z ≤ y
hyxulp : ulp n y ≤ 2 * x
ht : t = round (y - round (y - z))
hs₁ : x < t → s = round (x + round (t - x))
hs₂ : t ≤ x → s = round (x - round (x - t))
hxw : x ≤ w
hwy : w ≤ y
hxy : x ≤ y
ht' : t = y - round (y - z)
htx' : t ≤ y - round (y - x)
hzy' : round (y - z) ≤ y
hft : trunc n t = t
hfx' : round (y - round (y - x)) = y - round (y - x)
hfut : ∀ {u : ℕ}, x ≤ u → u < t → trunc n u = u → u ≤ round (y - round (y - x)) → trunc n (t - u) = t - u
⊢ trunc n (w - s) = w - s ∧ trunc n (s - w) = s - w