let h, x be Real; :: thesis: for f being Function of REAL,REAL holds (cD ((cD (f,h)),h)) . x = ((fD (f,h)) . x) - ((bD (f,h)) . x)
let f be Function of REAL,REAL; :: thesis: (cD ((cD (f,h)),h)) . x = ((fD (f,h)) . x) - ((bD (f,h)) . x)
(cD ((cD (f,h)),h)) . x = ((cD (f,h)) . (x + (h / 2))) - ((cD (f,h)) . (x - (h / 2))) by DIFF_1:5
.= ((f . ((x + (h / 2)) + (h / 2))) - (f . ((x + (h / 2)) - (h / 2)))) - ((cD (f,h)) . (x - (h / 2))) by DIFF_1:5
.= ((f . ((x + (h / 2)) + (h / 2))) - (f . ((x + (h / 2)) - (h / 2)))) - ((f . ((x - (h / 2)) + (h / 2))) - (f . ((x - (h / 2)) - (h / 2)))) by DIFF_1:5
.= ((f . (x + h)) - (f . x)) - ((f . x) - (f . (x - h)))
.= ((fD (f,h)) . x) - ((f . x) - (f . (x - h))) by DIFF_1:3
.= ((fD (f,h)) . x) - ((bD (f,h)) . x) by DIFF_1:4 ;
hence (cD ((cD (f,h)),h)) . x = ((fD (f,h)) . x) - ((bD (f,h)) . x) ; :: thesis: verum