theorem Th39: :: INTEGR11:39

for Z being open Subset of REAL st Z c= dom (tan - sec) & ( for x being Real st x in Z holds
( 1 + (sin . x) <> 0 & 1 - (sin . x) <> 0 ) ) holds
( tan - sec is_differentiable_on Z & ( for x being Real st x in Z holds
((tan - sec) `| Z) . x = 1 / (1 + (sin . x)) ) )