theorem extracted_formal_statement_5.{u_2} {ι : Type u_2} [inst : Nonempty ι] {g h : ι → ℝ} (hg : BddAbove (Set.range g))
  (hh : BddAbove (Set.range h)) (h_original_goal : (⨆ i, g i) + ⨆ i, h i = ⨆ i, ⨆ j, g i + h j) :
  ⨆ i, g i + ⨆ i, h i = ⨆ i, ⨆ j, g i + h j := sorry

State before:
case h.e'_4
ι : Type u_2
inst✝ : Nonempty ι
g h : ι → ℝ
hg : BddAbove (Set.range g)
hh : BddAbove (Set.range h)
⊢ (⨆ i, g i) + ⨆ i, h i = ⨆ i, ⨆ j, g i + h j

Tactic: rw [ciSup_add hg]

State after:
[case h.e'_4
 ι : Type u_2
 inst✝ : Nonempty ι
 g h : ι → ℝ
 hg : BddAbove (Set.range g)
 hh : BddAbove (Set.range h)
 ⊢ ⨆ i, g i + ⨆ i, h i = ⨆ i, ⨆ j, g i + h j]

Relevant lemmas:
ciSup_add : ∀ {ι : Type u_1} {G : Type u_2} [inst : AddGroup G] [inst_1 : ConditionallyCompleteLattice G]
  [inst_2 : Nonempty ι] {f : ι → G} [inst_3 : AddRightMono G],
  BddAbove (Set.range f) → ∀ (a : G), (⨆ i, f i) + a = ⨆ i, f i + a