theorem :: PDIFF_4:25

for f being PartFunc of (REAL 3),REAL
for u0 being Element of REAL 3
for N being Neighbourhood of (proj (1,3)) . u0 st f is_partial_differentiable_in u0,1 & N c= dom (SVF1 (1,f,u0)) holds
for h being non-zero 0 -convergent Real_Sequence
for c being constant Real_Sequence st rng c = {((proj (1,3)) . u0)} & rng (h + c) c= N holds
( (h ") (#) (((SVF1 (1,f,u0)) /* (h + c)) - ((SVF1 (1,f,u0)) /* c)) is convergent & partdiff (f,u0,1) = lim ((h ") (#) (((SVF1 (1,f,u0)) /* (h + c)) - ((SVF1 (1,f,u0)) /* c))) )