∀ {Γ : Subgroup (Matrix.SpecialLinearGroup (Fin 2) ℤ)} {g : GL (Fin 2) ℝ} {x : Matrix.SpecialLinearGroup (Fin 2) ℤ},
  (x ∈ CongruenceSubgroup.conjGL Γ g) =
    ∃ y ∈ Γ,
      Matrix.SpecialLinearGroup.toGL ((Matrix.SpecialLinearGroup.map (Int.castRingHom ℝ)) y) =
        g * Matrix.SpecialLinearGroup.toGL ((Matrix.SpecialLinearGroup.map (Int.castRingHom ℝ)) x) * g⁻¹