/-- If `E` is a fiber bundle over a conditionally complete linear order,
then it is trivial over any closed interval. -/
theorem FiberBundle.exists_trivialization_Icc_subset [ConditionallyCompleteLinearOrder B]
    [OrderTopology B] [FiberBundle F E] (a b : B) :
    ∃ e : Trivialization F (π F E), Icc a b ⊆ e.baseSet := by
  obtain ⟨ea, hea⟩ : ∃ ea : Trivialization F (π F E), a ∈ ea.baseSet :=
    ⟨trivializationAt F E a, mem_baseSet_trivializationAt F E a⟩
  -- If `a < b`, then `[a, b] = ∅`, and the statement is trivial
  cases' lt_or_le b a with hab hab
  · exact ⟨ea, by simp [*]⟩
  /- Let `s` be the set of points `x ∈ [a, b]` such that `E` is trivializable over `[a, x]`.
    We need to show that `b ∈ s`. Let `c = Sup s`. We will show that `c ∈ s` and `c = b`. -/
  set s : Set B := { x ∈ Icc a b | ∃ e : Trivialization F (π F E), Icc a x ⊆ e.baseSet }
  have ha : a ∈ s := ⟨left_mem_Icc.2 hab, ea, by simp [hea]⟩
  have sne : s.Nonempty := ⟨a, ha⟩
  have hsb : b ∈ upperBounds s := fun x hx => hx.1.2
  have sbd : BddAbove s := ⟨b, hsb⟩
  set c := sSup s
  have hsc : IsLUB s c := isLUB_csSup sne sbd
  have hc : c ∈ Icc a b := ⟨hsc.1 ha, hsc.2 hsb⟩
  obtain ⟨-, ec : Trivialization F (π F E), hec : Icc a c ⊆ ec.baseSet⟩ : c ∈ s := by
    rcases hc.1.eq_or_lt with heq | hlt
    · rwa [← heq]
    refine ⟨hc, ?_⟩
    /- In order to show that `c ∈ s`, consider a trivialization `ec` of `proj` over a neighborhood
      of `c`. Its base set includes `(c', c]` for some `c' ∈ [a, c)`. -/
    obtain ⟨ec, hc⟩ : ∃ ec : Trivialization F (π F E), c ∈ ec.baseSet :=
      ⟨trivializationAt F E c, mem_baseSet_trivializationAt F E c⟩
    obtain ⟨c', hc', hc'e⟩ : ∃ c' ∈ Ico a c, Ioc c' c ⊆ ec.baseSet :=
      (mem_nhdsWithin_Iic_iff_exists_mem_Ico_Ioc_subset hlt).1
        (mem_nhdsWithin_of_mem_nhds <| IsOpen.mem_nhds ec.open_baseSet hc)
    /- Since `c' < c = Sup s`, there exists `d ∈ s ∩ (c', c]`. Let `ead` be a trivialization of
      `proj` over `[a, d]`. Then we can glue `ead` and `ec` into a trivialization over `[a, c]`. -/
    obtain ⟨d, ⟨hdab, ead, had⟩, hd⟩ : ∃ d ∈ s, d ∈ Ioc c' c := hsc.exists_between hc'.2
    refine' ⟨ead.piecewiseLe ec d (had ⟨hdab.1, le_rfl⟩) (hc'e hd), subset_ite.2 _⟩
    refine' ⟨fun x hx => had ⟨hx.1.1, hx.2⟩, fun x hx => hc'e ⟨hd.1.trans (not_le.1 hx.2), hx.1.2⟩⟩
  /- So, `c ∈ s`. Let `ec` be a trivialization of `proj` over `[a, c]`.  If `c = b`, then we are
    done. Otherwise we show that `proj` can be trivialized over a larger interval `[a, d]`,
    `d ∈ (c, b]`, hence `c` is not an upper bound of `s`. -/
  rcases hc.2.eq_or_lt with heq | hlt
  · exact ⟨ec, heq ▸ hec⟩
  rsuffices ⟨d, hdcb, hd⟩ : ∃ d ∈ Ioc c b, ∃ e : Trivialization F (π F E), Icc a d ⊆ e.baseSet
  · exact ((hsc.1 ⟨⟨hc.1.trans hdcb.1.le, hdcb.2⟩, hd⟩).not_lt hdcb.1).elim
  /- Since the base set of `ec` is open, it includes `[c, d)` (hence, `[a, d)`) for some
    `d ∈ (c, b]`. -/
  obtain ⟨d, hdcb, hd⟩ : ∃ d ∈ Ioc c b, Ico c d ⊆ ec.baseSet :=
    (mem_nhdsWithin_Ici_iff_exists_mem_Ioc_Ico_subset hlt).1
      (mem_nhdsWithin_of_mem_nhds <| IsOpen.mem_nhds ec.open_baseSet (hec ⟨hc.1, le_rfl⟩))
  have had : Ico a d ⊆ ec.baseSet := Ico_subset_Icc_union_Ico.trans (union_subset hec hd)
  by_cases he : Disjoint (Iio d) (Ioi c)
  · /- If `(c, d) = ∅`, then let `ed` be a trivialization of `proj` over a neighborhood of `d`.
      Then the disjoint union of `ec` restricted to `(-∞, d)` and `ed` restricted to `(c, ∞)` is
      a trivialization over `[a, d]`. -/
    obtain ⟨ed, hed⟩ : ∃ ed : Trivialization F (π F E), d ∈ ed.baseSet :=
      ⟨trivializationAt F E d, mem_baseSet_trivializationAt F E d⟩
    refine' ⟨d, hdcb,
      (ec.restrOpen (Iio d) isOpen_Iio).disjointUnion (ed.restrOpen (Ioi c) isOpen_Ioi)
        (he.mono (inter_subset_right _ _) (inter_subset_right _ _)), fun x hx => _⟩
    rcases hx.2.eq_or_lt with (rfl | hxd)
    

Proof state:
case pos.intro.inl
ι : Type u_1
B : Type u_2
F : Type u_3
X : Type u_4
inst✝⁷ : TopologicalSpace X
inst✝⁶ : TopologicalSpace B
inst✝⁵ : TopologicalSpace F
E : B → Type u_5
inst✝⁴ : TopologicalSpace (TotalSpace F E)
inst✝³ : (b : B) → TopologicalSpace (E b)
inst✝² : ConditionallyCompleteLinearOrder B
inst✝¹ : OrderTopology B
inst✝ : FiberBundle F E
a b : B
ea : Trivialization F TotalSpace.proj
hea : a ∈ ea.baseSet
hab : a ≤ b
s : Set B := {x | x ∈ Icc a b ∧ ∃ e, Icc a x ⊆ e.baseSet}
ha : a ∈ s
sne : Set.Nonempty s
hsb : b ∈ upperBounds s
sbd : BddAbove s
c : B := sSup s
hsc : IsLUB s c
hc : c ∈ Icc a b
ec : Trivialization F TotalSpace.proj
hec : Icc a c ⊆ ec.baseSet
hlt : c < b
ed : Trivialization F TotalSpace.proj
x : B
hdcb : x ∈ Ioc c b
hd : Ico c x ⊆ ec.baseSet
had : Ico a x ⊆ ec.baseSet
he : Disjoint (Iio x) (Ioi c)
hed : x ∈ ed.baseSet
hx : x ∈ Icc a x
⊢ x ∈
    (Trivialization.disjointUnion (Trivialization.restrOpen ec (Iio x) (_ : IsOpen (Iio x)))
        (Trivialization.restrOpen ed (Ioi c) (_ : IsOpen (Ioi c)))
        (_ :
          Disjoint (Trivialization.restrOpen ec (Iio x) (_ : IsOpen (Iio x))).baseSet
            (Trivialization.restrOpen ed (Ioi c) (_ : IsOpen (Ioi c))).baseSet)).baseSet
case pos.intro.inr
ι : Type u_1
B : Type u_2
F : Type u_3
X : Type u_4
inst✝⁷ : TopologicalSpace X
inst✝⁶ : TopologicalSpace B
inst✝⁵ : TopologicalSpace F
E : B → Type u_5
inst✝⁴ : TopologicalSpace (TotalSpace F E)
inst✝³ : (b : B) → TopologicalSpace (E b)
inst✝² : ConditionallyCompleteLinearOrder B
inst✝¹ : OrderTopology B
inst✝ : FiberBundle F E
a b : B
ea : Trivialization F TotalSpace.proj
hea : a ∈ ea.baseSet
hab : a ≤ b
s : Set B := {x | x ∈ Icc a b ∧ ∃ e, Icc a x ⊆ e.baseSet}
ha : a ∈ s
sne : Set.Nonempty s
hsb : b ∈ upperBounds s
sbd : BddAbove s
c : B := sSup s
hsc : IsLUB s c
hc : c ∈ Icc a b
ec : Trivialization F TotalSpace.proj
hec : Icc a c ⊆ ec.baseSet
hlt : c < b
d : B
hdcb : d ∈ Ioc c b
hd : Ico c d ⊆ ec.baseSet
had : Ico a d ⊆ ec.baseSet
he : Disjoint (Iio d) (Ioi c)
ed : Trivialization F TotalSpace.proj
hed : d ∈ ed.baseSet
x : B
hx : x ∈ Icc a d
hxd : x < d
⊢ x ∈
    (Trivialization.disjointUnion (Trivialization.restrOpen ec (Iio d) (_ : IsOpen (Iio d)))
        (Trivialization.restrOpen ed (Ioi c) (_ : IsOpen (Ioi c)))
        (_ :
          Disjoint (Trivialization.restrOpen ec (Iio d) (_ : IsOpen (Iio d))).baseSet
            (Trivialization.restrOpen ed (Ioi c) (_ : IsOpen (Ioi c))).baseSet)).baseSet

Next tactic:
exacts [Or.inr ⟨hed, hdcb.1⟩, Or.inl ⟨had ⟨hx.1, hxd⟩, hxd⟩]