theorem Th12: :: INTEGR11:12

for a, b being Real
for n being Element of NAT st a * (n + 1) <> 0 holds
( (1 / (a * (n + 1))) (#) ((#Z (n + 1)) * (AffineMap (a,b))) is_differentiable_on REAL & ( for x being Real holds (((1 / (a * (n + 1))) (#) ((#Z (n + 1)) * (AffineMap (a,b)))) `| REAL) . x = ((a * x) + b) #Z n ) )