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