theorem Th12: :: FDIFF_2:12

for x0, g being Real
for f being PartFunc of REAL,REAL holds
( f is_differentiable_in x0 & diff (f,x0) = g iff ( ex N being Neighbourhood of x0 st N c= dom f & ( for h being non-zero 0 -convergent Real_Sequence
for c being constant Real_Sequence st rng c = {x0} & rng (h + c) c= dom f holds
( (h ") (#) ((f /* (h + c)) - (f /* c)) is convergent & lim ((h ") (#) ((f /* (h + c)) - (f /* c))) = g ) ) ) )