apply Exists.elim σ.2
σ : Instantiation
⊢ ∃ σ_inv, ↑σ ∘ ↑σ_inv = id ∧ ↑σ_inv ∘ ↑σ = id
σ : Instantiation
⊢ ∀ (a : VarName → VarName), ↑σ ∘ a = id ∧ a ∘ ↑σ = id → ∃ σ_inv, ↑σ ∘ ↑σ_inv = id ∧ ↑σ_inv ∘ ↑σ = id