have hx1 : x.1 = 1 := by
  linarith
case mp
x : ℝ × ℝ
imp : ∀ (i : ℝ), 0 ≤ i → i ≤ 1 → (i ≤ x.1 ∧ 0 ≤ x.2) ∧ x.1 ≤ 1 ∧ x.2 ≤ i
h1 : (0 ≤ x.1 ∧ 0 ≤ x.2) ∧ x.1 ≤ 1 ∧ x.2 ≤ 0
h2 : (1 ≤ x.1 ∧ 0 ≤ x.2) ∧ x.1 ≤ 1 ∧ x.2 ≤ 1
⊢ x ∈ Set.singleton (1, 0)
case mp
x : ℝ × ℝ
imp : ∀ (i : ℝ), 0 ≤ i → i ≤ 1 → (i ≤ x.1 ∧ 0 ≤ x.2) ∧ x.1 ≤ 1 ∧ x.2 ≤ i
h1 : (0 ≤ x.1 ∧ 0 ≤ x.2) ∧ x.1 ≤ 1 ∧ x.2 ≤ 0
h2 : (1 ≤ x.1 ∧ 0 ≤ x.2) ∧ x.1 ≤ 1 ∧ x.2 ≤ 1
hx1 : x.1 = 1
⊢ x ∈ Set.singleton (1, 0)