(** * Taylor-Lagrange formula *)

Theorem Taylor_Lagrange :
forall f n x y, x < y ->
( forall t, x <= t <= y -> forall k, (k <= S n)%nat -> ex_derive_n f k t ) ->
exists zeta, x < zeta < y /\
f y =  sum_f_R0 (fun m => (y-x) ^ m / INR (fact m) * Derive_n f m x )  n
+ (y-x) ^ (S n) / INR (fact (S n)) * Derive_n f (S n) zeta.