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