theorem Th43: :: FDIFF_12:43

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