proposition: Lemma Ropp_add x y : -(x + y) == -x + -y.
proof: Proof. rewrite <- ((Radd_0_l Rth) (-(x+y))). rewrite <- ((Ropp_def Rth) x). rewrite <- ((Radd_0_l Rth) (x + - x + - (x + y))). rewrite <- ((Ropp_def Rth) y). rewrite ((Radd_comm Rth) x). rewrite ((Radd_comm Rth) y). rewrite <- ((Radd_assoc Rth) (-y)). rewrite <- ((Radd_assoc Rth) (- x)). rewrite ((Radd_assoc Rth) y). rewrite ((Radd_comm Rth) y). rewrite <- ((Radd_assoc Rth) (- x)). rewrite ((Radd_assoc Rth) y). rewrite ((Radd_comm Rth) y), (Ropp_def Rth). rewrite ((Radd_comm Rth) (-x) 0), (Radd_0_l Rth). now apply (Radd_comm Rth). Qed.
imports: Setoid Morphisms BinPos BinNat