let n be Element of NAT ; :: thesis: for r, h, x being Real
for f1, f2 being Function of REAL,REAL holds ((bdif ((f1 + (r (#) f2)),h)) . (n + 1)) . x = (((bdif (f1,h)) . (n + 1)) . x) + (r * (((bdif (f2,h)) . (n + 1)) . x))
let r, h, x be Real; :: thesis: for f1, f2 being Function of REAL,REAL holds ((bdif ((f1 + (r (#) f2)),h)) . (n + 1)) . x = (((bdif (f1,h)) . (n + 1)) . x) + (r * (((bdif (f2,h)) . (n + 1)) . x))
let f1, f2 be Function of REAL,REAL; :: thesis: ((bdif ((f1 + (r (#) f2)),h)) . (n + 1)) . x = (((bdif (f1,h)) . (n + 1)) . x) + (r * (((bdif (f2,h)) . (n + 1)) . x))
set g = r (#) f2;
((bdif ((f1 + (r (#) f2)),h)) . (n + 1)) . x = (((bdif (f1,h)) . (n + 1)) . x) + (((bdif ((r (#) f2),h)) . (n + 1)) . x) by DIFF_1:15
.= (((bdif (f1,h)) . (n + 1)) . x) + (r * (((bdif (f2,h)) . (n + 1)) . x)) by DIFF_1:14 ;
hence ((bdif ((f1 + (r (#) f2)),h)) . (n + 1)) . x = (((bdif (f1,h)) . (n + 1)) . x) + (r * (((bdif (f2,h)) . (n + 1)) . x)) ; :: thesis: verum