let h, x be Real; :: thesis: for f being Function of REAL,REAL holds ((bdif (f,h)) . 1) . x = (((bdif (f,h)) . 0) . x) - (((bdif (f,h)) . 0) . (x - h))
let f be Function of REAL,REAL; :: thesis: ((bdif (f,h)) . 1) . x = (((bdif (f,h)) . 0) . x) - (((bdif (f,h)) . 0) . (x - h))
((bdif (f,h)) . 1) . x = (bD (f,h)) . x by DIFF_3:11
.= (f . x) - (f . (x - h)) by DIFF_1:4
.= (((bdif (f,h)) . 0) . x) - (f . (x - h)) by DIFF_1:def 7
.= (((bdif (f,h)) . 0) . x) - (((bdif (f,h)) . 0) . (x - h)) by DIFF_1:def 7 ;
hence ((bdif (f,h)) . 1) . x = (((bdif (f,h)) . 0) . x) - (((bdif (f,h)) . 0) . (x - h)) ; :: thesis: verum