∀ {R : Type u_3} {M : Type u_4} {N : Type u_5} [inst : CommRing R] [inst_1 : AddCommGroup M] [inst_2 : AddCommGroup N]
  [inst_3 : Module R M] [inst_4 : Module R N] (toFun : M → N)
  (toFun_smul : ∀ (a : R) (x : M), toFun (a • x) = (a * a) • toFun x)
  (polar_add_left :
    ∀ (x x' y : M), QuadraticMap.polar toFun (x + x') y = QuadraticMap.polar toFun x y + QuadraticMap.polar toFun x' y)
  (polar_smul_left : ∀ (a : R) (x y : M), QuadraticMap.polar toFun (a • x) y = a • QuadraticMap.polar toFun x y)
  (a : M), (QuadraticMap.ofPolar toFun toFun_smul polar_add_left polar_smul_left) a = toFun a