let x, h be Real; :: thesis: for f being Function of REAL ,REAL holds [!f,(x - h),x!] = (((bdif f,h) . 1) . x) / h
let f be Function of REAL ,REAL ; :: thesis: [!f,(x - h),x!] = (((bdif f,h) . 1) . x) / h
[!f,(x - h),x!] = [!f,x,(x - h)!] by DIFF_1:29
.= ((bD f,h) . x) / h by DIFF_1:4
.= ((bD ((bdif f,h) . 0 ),h) . x) / h by DIFF_1:def 7
.= (((bdif f,h) . (0 + 1)) . x) / h by DIFF_1:def 7 ;
hence [!f,(x - h),x!] = (((bdif f,h) . 1) . x) / h ; :: thesis: verum