theorem :: PDIFF_3:20

for f being PartFunc of (REAL 2),REAL
for z0 being Element of REAL 2
for N being Neighbourhood of (proj (2,2)) . z0 st f is_hpartial_differentiable`22_in z0 & N c= dom (SVF1 (2,(pdiff1 (f,2)),z0)) holds
for h being non-zero 0 -convergent Real_Sequence
for c being constant Real_Sequence st rng c = {((proj (2,2)) . z0)} & rng (h + c) c= N holds
( (h ") (#) (((SVF1 (2,(pdiff1 (f,2)),z0)) /* (h + c)) - ((SVF1 (2,(pdiff1 (f,2)),z0)) /* c)) is convergent & hpartdiff22 (f,z0) = lim ((h ") (#) (((SVF1 (2,(pdiff1 (f,2)),z0)) /* (h + c)) - ((SVF1 (2,(pdiff1 (f,2)),z0)) /* c))) )