theorem Th7: :: FDIFF_2:7

for g being Real
for f being PartFunc of REAL,REAL st ( for h being non-zero 0 -convergent Real_Sequence
for c being constant Real_Sequence st rng c = {g} & rng (h + c) c= dom f & {g} c= dom f holds
(h ") (#) ((f /* (h + c)) - (f /* c)) is convergent ) holds
for h1, h2 being non-zero 0 -convergent Real_Sequence
for c being constant Real_Sequence st rng c = {g} & rng (h1 + c) c= dom f & rng (h2 + c) c= dom f & {g} c= dom f holds
lim ((h1 ") (#) ((f /* (h1 + c)) - (f /* c))) = lim ((h2 ") (#) ((f /* (h2 + c)) - (f /* c)))