theorem Th9: :: INTEGR13:9

for Z being open Subset of REAL st Z c= dom ((1 / 2) (#) ((sin - cos) / exp_R)) holds
( (1 / 2) (#) ((sin - cos) / exp_R) is_differentiable_on Z & ( for x being Real st x in Z holds
(((1 / 2) (#) ((sin - cos) / exp_R)) `| Z) . x = (cos . x) / (exp_R . x) ) )