/-- The sum of two lower semicontinuous functions is lower semicontinuous. Formulated with an
explicit continuity assumption on addition, for application to `EReal`. The unprimed version of
the lemma uses `[ContinuousAdd]`. -/
theorem LowerSemicontinuousWithinAt.add' {f g : α → γ} (hf : LowerSemicontinuousWithinAt f s x)
    (hg : LowerSemicontinuousWithinAt g s x)
    (hcont : ContinuousAt (fun p : γ × γ => p.1 + p.2) (f x, g x)) :
    LowerSemicontinuousWithinAt (fun z => f z + g z) s x := by
  intro y hy
  obtain ⟨u, v, u_open, xu, v_open, xv, h⟩ :
    ∃ u v : Set γ,
      IsOpen u ∧ f x ∈ u ∧ IsOpen v ∧ g x ∈ v ∧ u ×ˢ v ⊆ { p : γ × γ | y < p.fst + p.snd } :=
    mem_nhds_prod_iff'.1 (hcont (isOpen_Ioi.mem_nhds hy))
  by_cases hx₁ : ∃ l, l < f x
  · obtain ⟨z₁, z₁lt, h₁⟩ : ∃ z₁ < f x, Ioc z₁ (f x) ⊆ u :=
      exists_Ioc_subset_of_mem_nhds (u_open.mem_nhds xu) hx₁
    by_cases hx₂ : ∃ l, l < g x
    · obtain ⟨z₂, z₂lt, h₂⟩ : ∃ z₂ < g x, Ioc z₂ (g x) ⊆ v :=
        exists_Ioc_subset_of_mem_nhds (v_open.mem_nhds xv) hx₂
      filter_upwards [hf z₁ z₁lt, hg z₂ z₂lt] with z h₁z h₂z
      have A1 : min (f z) (f x) ∈ u := by
        by_cases H : f z ≤ f x
        · simp [H]
          exact h₁ ⟨h₁z, H⟩
        · simp [le_of_not_le H]
          exact h₁ ⟨z₁lt, le_rfl⟩
      have A2 : min (g z) (g x) ∈ v := by
        by_cases H : g z ≤ g x
        · simp [H]
          exact h₂ ⟨h₂z, H⟩
        · simp [le_of_not_le H]
          exact h₂ ⟨z₂lt, le_rfl⟩
      have : (min (f z) (f x), min (g z) (g x)) ∈ u ×ˢ v := ⟨A1, A2⟩
      calc
        y < min (f z) (f x) + min (g z) (g x) := h this
        _ ≤ f z + g z := add_le_add (min_le_left _ _) (min_le_left _ _)

    · simp only [not_exists, not_lt] at hx₂
      filter_upwards [hf z₁ z₁lt] with z h₁z
      have A1 : min (f z) (f x) ∈ u := by
        by_cases H : f z ≤ f x
        · simp [H]
          exact h₁ ⟨h₁z, H⟩
        · simp [le_of_not_le H]
          exact h₁ ⟨z₁lt, le_rfl⟩
      have : (min (f z) (f x), g x) ∈ u ×ˢ v := ⟨A1, xv⟩
      calc
        y < min (f z) (f x) + g x := h this
        _ ≤ f z + g z := add_le_add (min_le_left _ _) (hx₂ (g z))

  · simp only [not_exists, not_lt] at hx₁
    by_cases hx₂ : ∃ l, l < g x
    · obtain ⟨z₂, z₂lt, h₂⟩ : ∃ z₂ < g x, Ioc z₂ (g x) ⊆ v :=
        exists_Ioc_subset_of_mem_nhds (v_open.mem_nhds xv) hx₂
      filter_upwards [hg z₂ z₂lt] with z h₂z
      have A2 : min (g z) (g x) ∈ v := by
        by_cases H : g z ≤ g x
        · simp [H]
          exact h₂ ⟨h₂z, H⟩
        · simp [le_of_not_le H]
          exact h₂ ⟨z₂lt, le_rfl⟩
      

Proof state:
case h
α : Type u_1
inst✝⁴ : TopologicalSpace α
β : Type u_2
inst✝³ : Preorder β
f✝ g✝ : α → β
x : α
s t : Set α
y✝ z✝ : β
ι : Type u_3
γ : Type u_4
inst✝² : LinearOrderedAddCommMonoid γ
inst✝¹ : TopologicalSpace γ
inst✝ : OrderTopology γ
f g : α → γ
hf : LowerSemicontinuousWithinAt f s x
hg : LowerSemicontinuousWithinAt g s x
hcont : ContinuousAt (fun p => p.1 + p.2) (f x, g x)
y : γ
hy : y < (fun z => f z + g z) x
u v : Set γ
u_open : IsOpen u
xu : f x ∈ u
v_open : IsOpen v
xv : g x ∈ v
h : u ×ˢ v ⊆ {p | y < p.1 + p.2}
hx₁ : ∀ (x_1 : γ), f x ≤ x_1
hx₂ : ∃ l, l < g x
z₂ : γ
z₂lt : z₂ < g x
h₂ : Ioc z₂ (g x) ⊆ v
z : α
h₂z : z₂ < g z
A2 : min (g z) (g x) ∈ v
⊢ y < f z + g z

Next tactic:
have : (f x, min (g z) (g x)) ∈ u ×ˢ v := ⟨xu, A2⟩