let h, x be Real; :: thesis: for f being Function of REAL,REAL holds [!f,x,(x + h)!] = (((fdif (f,h)) . 1) . x) / h
let f be Function of REAL,REAL; :: thesis: [!f,x,(x + h)!] = (((fdif (f,h)) . 1) . x) / h
[!f,x,(x + h)!] = [!f,(x + h),x!] by DIFF_1:29
.= ((fD (f,h)) . x) / h by DIFF_1:3
.= ((fD (((fdif (f,h)) . 0),h)) . x) / h by DIFF_1:def 6
.= (((fdif (f,h)) . (0 + 1)) . x) / h by DIFF_1:def 6 ;
hence [!f,x,(x + h)!] = (((fdif (f,h)) . 1) . x) / h ; :: thesis: verum