theorem :: PDIFF_5:41

for f being PartFunc of (REAL 3),REAL
for u0 being Element of REAL 3
for N being Neighbourhood of (proj (2,3)) . u0 st f is_hpartial_differentiable`22_in u0 & N c= dom (SVF1 (2,(pdiff1 (f,2)),u0)) holds
for h being non-zero 0 -convergent Real_Sequence
for c being constant Real_Sequence st rng c = {((proj (2,3)) . u0)} & rng (h + c) c= N holds
( (h ") (#) (((SVF1 (2,(pdiff1 (f,2)),u0)) /* (h + c)) - ((SVF1 (2,(pdiff1 (f,2)),u0)) /* c)) is convergent & hpartdiff22 (f,u0) = lim ((h ") (#) (((SVF1 (2,(pdiff1 (f,2)),u0)) /* (h + c)) - ((SVF1 (2,(pdiff1 (f,2)),u0)) /* c))) )