/-- The left chart for the topological space `[x, y]`, defined on `[x,y)` and sending `x` to `0` in
`EuclideanHalfSpace 1`.
-/
def IccLeftChart (x y : ℝ) [h : Fact (x < y)] :
    PartialHomeomorph (Icc x y) (EuclideanHalfSpace 1) where
  source := { z : Icc x y | z.val < y }
  target := { z : EuclideanHalfSpace 1 | z.val 0 < y - x }
  toFun := fun z : Icc x y => ⟨fun _ => z.val - x, sub_nonneg.mpr z.property.1⟩
  invFun z := ⟨min (z.val 0 + x) y, by simp [le_refl, z.prop, le_of_lt h.out]⟩
  map_source' := by simp only [imp_self, sub_lt_sub_iff_right, mem_setOf_eq, forall_true_iff]
  map_target' := by
    simp only [min_lt_iff, mem_setOf_eq]; intro z hz; left
    linarith
  left_inv' := by
    rintro ⟨z, hz⟩ h'z
    simp only [mem_setOf_eq, mem_Icc] at hz h'z
    simp only [hz, min_eq_left, sub_add_cancel]
  right_inv' := by
    rintro ⟨z, hz⟩ h'z
    rw [Subtype.mk_eq_mk]
    funext i
    dsimp at hz h'z
    have A : x + z 0 ≤ y := by linarith
    rw [Subsingleton.elim i 0]
    simp only [A, add_comm, add_sub_cancel', min_eq_left]
  open_source :=
    haveI : IsOpen { z : ℝ | z < y } := isOpen_Iio
    this.preimage continuous_subtype_val
  open_target := by
    have : IsOpen { z : ℝ | z < y - x } := isOpen_Iio
    have : IsOpen { z : EuclideanSpace ℝ (Fin 1) | z 0 < y - x } :=
      this.preimage (@continuous_apply (Fin 1) (fun _ => ℝ) _ 0)
    exact this.preimage continuous_subtype_val
  continuousOn_toFun := by
    apply Continuous.continuousOn
    apply Continuous.subtype_mk
    have : Continuous fun (z : ℝ) (_ : Fin 1) => z - x :=
      Continuous.sub (continuous_pi fun _ => continuous_id) continuous_const
    exact this.comp continuous_subtype_val
  continuousOn_invFun := by
    

Proof state:
x y : ℝ
h : Fact (x < y)
⊢ ContinuousOn
    { toFun := fun z => { val := fun x_1 => ↑z - x, property := (_ : 0 ≤ ↑z - x) },
        invFun := fun z => { val := min (↑z 0 + x) y, property := (_ : min (↑z 0 + x) y ∈ Icc x y) },
        source := {z | ↑z < y}, target := {z | ↑z 0 < y - x},
        map_source' := (_ : ∀ (a : ↑(Icc x y)), ↑a < y → ↑a - x < y - x),
        map_target' :=
          (_ :
            ∀ ⦃x_1 : EuclideanHalfSpace 1⦄,
              x_1 ∈ {z | ↑z 0 < y - x} →
                (fun z => { val := min (↑z 0 + x) y, property := (_ : min (↑z 0 + x) y ∈ Icc x y) }) x_1 ∈
                  {z | ↑z < y}),
        left_inv' :=
          (_ :
            ∀ ⦃x_1 : ↑(Icc x y)⦄,
              x_1 ∈ {z | ↑z < y} →
                (fun z => { val := min (↑z 0 + x) y, property := (_ : min (↑z 0 + x) y ∈ Icc x y) })
                    ((fun z => { val := fun x_2 => ↑z - x, property := (_ : 0 ≤ ↑z - x) }) x_1) =
                  x_1),
        right_inv' :=
          (_ :
            ∀ ⦃x_1 : EuclideanHalfSpace 1⦄,
              x_1 ∈ {z | ↑z 0 < y - x} →
                (fun z => { val := fun x_2 => ↑z - x, property := (_ : 0 ≤ ↑z - x) })
                    ((fun z => { val := min (↑z 0 + x) y, property := (_ : min (↑z 0 + x) y ∈ Icc x y) }) x_1) =
                  x_1) }.invFun
    { toFun := fun z => { val := fun x_1 => ↑z - x, property := (_ : 0 ≤ ↑z - x) },
        invFun := fun z => { val := min (↑z 0 + x) y, property := (_ : min (↑z 0 + x) y ∈ Icc x y) },
        source := {z | ↑z < y}, target := {z | ↑z 0 < y - x},
        map_source' := (_ : ∀ (a : ↑(Icc x y)), ↑a < y → ↑a - x < y - x),
        map_target' :=
          (_ :
            ∀ ⦃x_1 : EuclideanHalfSpace 1⦄,
              x_1 ∈ {z | ↑z 0 < y - x} →
                (fun z => { val := min (↑z 0 + x) y, property := (_ : min (↑z 0 + x) y ∈ Icc x y) }) x_1 ∈
                  {z | ↑z < y}),
        left_inv' :=
          (_ :
            ∀ ⦃x_1 : ↑(Icc x y)⦄,
              x_1 ∈ {z | ↑z < y} →
                (fun z => { val := min (↑z 0 + x) y, property := (_ : min (↑z 0 + x) y ∈ Icc x y) })
                    ((fun z => { val := fun x_2 => ↑z - x, property := (_ : 0 ≤ ↑z - x) }) x_1) =
                  x_1),
        right_inv' :=
          (_ :
            ∀ ⦃x_1 : EuclideanHalfSpace 1⦄,
              x_1 ∈ {z | ↑z 0 < y - x} →
                (fun z => { val := fun x_2 => ↑z - x, property := (_ : 0 ≤ ↑z - x) })
                    ((fun z => { val := min (↑z 0 + x) y, property := (_ : min (↑z 0 + x) y ∈ Icc x y) }) x_1) =
                  x_1) }.target

Next tactic:
apply Continuous.continuousOn