theorem extracted_formal_statement_47.{u_5, u_4} {α : Type u_4} {β : Type u_5} [inst : NormedAddCommGroup α]
  [inst_1 : InnerProductSpace ℝ α] [inst_2 : NormedAddCommGroup β] [inst_3 : InnerProductSpace ℝ β] (A : α →L[ℝ] β)
  (a₁ a₂ : α) (h_original_goal : ‖A (a₁ - a₂)‖ = ‖A (a₂ - a₁)‖) : ‖-(A a₂ - A a₁)‖ = ‖A a₂ - A a₁‖ := sorry

State before:
α : Type u_4
β : Type u_5
inst✝³ : NormedAddCommGroup α
inst✝² : InnerProductSpace ℝ α
inst✝¹ : NormedAddCommGroup β
inst✝ : InnerProductSpace ℝ β
A : α →L[ℝ] β
a₁ a₂ : α
⊢ ‖A (a₁ - a₂)‖ = ‖A (a₂ - a₁)‖

Tactic: rw [map_sub, map_sub, ← neg_sub (A a₂) (A a₁)]

State after:
[α : Type u_4
 β : Type u_5
 inst✝³ : NormedAddCommGroup α
 inst✝² : InnerProductSpace ℝ α
 inst✝¹ : NormedAddCommGroup β
 inst✝ : InnerProductSpace ℝ β
 A : α →L[ℝ] β
 a₁ a₂ : α
 ⊢ ‖-(A a₂ - A a₁)‖ = ‖A a₂ - A a₁‖]

Relevant lemmas:
neg_sub : ∀ {α : Type u_1} [inst : SubtractionMonoid α] (a b : α), -(a - b) = b - a
map_sub : ∀ {G : Type u_7} {H : Type u_8} {F : Type u_9} [inst : FunLike F G H] [inst_4 : AddGroup G]
  [inst_5 : SubtractionMonoid H] [inst_6 : AddMonoidHomClass F G H] (f : F) (a b : G), f (a - b) = f a - f b