theorem Th1: :: FDIFF_12:1

for A, B being open Subset of REAL
for f1, f2 being PartFunc of REAL,REAL st f1 is_differentiable_on A & rng (f1 | A) c= B & f2 is_differentiable_on B holds
( f2 * f1 is_differentiable_on A & (f2 * f1) `| A = ((f2 `| B) * f1) (#) (f1 `| A) )