theorem Th47: :: FDIFF_12:47

for x0 being Real
for f1, f2 being PartFunc of REAL,REAL st f1 is_differentiable_in x0 & f2 is_left_differentiable_in f1 . x0 & ( for r1 being Real st r1 > 0 holds
ex r0 being Real st
( r0 > 0 & [.(x0 - r0),(x0 + r0).] c= dom ([.((f1 . x0) - r1),(f1 . x0).] |` f1) ) ) holds
( f2 * f1 is_differentiable_in x0 & diff ((f2 * f1),x0) = (Ldiff (f2,(f1 . x0))) * (diff (f1,x0)) )