/-- The right chart for the topological space `[x, y]`, defined on `(x,y]` and sending `y` to `0` in
`EuclideanHalfSpace 1`.
-/
def IccRightChart (x y : ℝ) [h : Fact (x < y)] :
    PartialHomeomorph (Icc x y) (EuclideanHalfSpace 1) where
  source := { z : Icc x y | x < z.val }
  target := { z : EuclideanHalfSpace 1 | z.val 0 < y - x }
  toFun z := ⟨fun _ => y - z.val, sub_nonneg.mpr z.property.2⟩
  invFun z :=
    ⟨max (y - z.val 0) x, by simp [le_refl, z.prop, le_of_lt h.out, sub_eq_add_neg]⟩
  map_source' := by simp only [imp_self, mem_setOf_eq, sub_lt_sub_iff_left, forall_true_iff]
  map_target' := by
    simp only [lt_max_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, sub_eq_add_neg, max_eq_left, add_add_neg_cancel'_right, neg_add_rev, neg_neg]
  right_inv' := by
    rintro ⟨z, hz⟩ h'z
    rw [Subtype.mk_eq_mk]
    funext i
    dsimp at hz h'z
    have A : x ≤ y - z 0 := by linarith
    rw [Subsingleton.elim i 0]
    simp only [A, sub_sub_cancel, max_eq_left]
  open_source :=
    haveI : IsOpen { z : ℝ | x < z } := isOpen_Ioi
    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) => y - z :=
      continuous_const.sub (continuous_pi fun _ => continuous_id)
    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 => y - ↑z, property := (_ : 0 ≤ y - ↑z) },
        invFun := fun z => { val := max (y - ↑z 0) x, property := (_ : max (y - ↑z 0) x ∈ Icc x y) },
        source := {z | x < ↑z}, target := {z | ↑z 0 < y - x},
        map_source' := (_ : ∀ (a : ↑(Icc x y)), x < ↑a → y - ↑a < y - x),
        map_target' :=
          (_ :
            ∀ ⦃x_1 : EuclideanHalfSpace 1⦄,
              x_1 ∈ {z | ↑z 0 < y - x} →
                (fun z => { val := max (y - ↑z 0) x, property := (_ : max (y - ↑z 0) x ∈ Icc x y) }) x_1 ∈
                  {z | x < ↑z}),
        left_inv' :=
          (_ :
            ∀ ⦃x_1 : ↑(Icc x y)⦄,
              x_1 ∈ {z | x < ↑z} →
                (fun z => { val := max (y - ↑z 0) x, property := (_ : max (y - ↑z 0) x ∈ Icc x y) })
                    ((fun z => { val := fun x_2 => y - ↑z, property := (_ : 0 ≤ y - ↑z) }) x_1) =
                  x_1),
        right_inv' :=
          (_ :
            ∀ ⦃x_1 : EuclideanHalfSpace 1⦄,
              x_1 ∈ {z | ↑z 0 < y - x} →
                (fun z => { val := fun x_2 => y - ↑z, property := (_ : 0 ≤ y - ↑z) })
                    ((fun z => { val := max (y - ↑z 0) x, property := (_ : max (y - ↑z 0) x ∈ Icc x y) }) x_1) =
                  x_1) }.invFun
    { toFun := fun z => { val := fun x_1 => y - ↑z, property := (_ : 0 ≤ y - ↑z) },
        invFun := fun z => { val := max (y - ↑z 0) x, property := (_ : max (y - ↑z 0) x ∈ Icc x y) },
        source := {z | x < ↑z}, target := {z | ↑z 0 < y - x},
        map_source' := (_ : ∀ (a : ↑(Icc x y)), x < ↑a → y - ↑a < y - x),
        map_target' :=
          (_ :
            ∀ ⦃x_1 : EuclideanHalfSpace 1⦄,
              x_1 ∈ {z | ↑z 0 < y - x} →
                (fun z => { val := max (y - ↑z 0) x, property := (_ : max (y - ↑z 0) x ∈ Icc x y) }) x_1 ∈
                  {z | x < ↑z}),
        left_inv' :=
          (_ :
            ∀ ⦃x_1 : ↑(Icc x y)⦄,
              x_1 ∈ {z | x < ↑z} →
                (fun z => { val := max (y - ↑z 0) x, property := (_ : max (y - ↑z 0) x ∈ Icc x y) })
                    ((fun z => { val := fun x_2 => y - ↑z, property := (_ : 0 ≤ y - ↑z) }) x_1) =
                  x_1),
        right_inv' :=
          (_ :
            ∀ ⦃x_1 : EuclideanHalfSpace 1⦄,
              x_1 ∈ {z | ↑z 0 < y - x} →
                (fun z => { val := fun x_2 => y - ↑z, property := (_ : 0 ≤ y - ↑z) })
                    ((fun z => { val := max (y - ↑z 0) x, property := (_ : max (y - ↑z 0) x ∈ Icc x y) }) x_1) =
                  x_1) }.target

Next tactic:
apply Continuous.continuousOn