theorem :: PDIFF_4:11

for x0, y0, z0 being Real
for u being Element of REAL 3
for f being PartFunc of (REAL 3),REAL st u = <*x0,y0,z0*> & f is_partial_differentiable_in u,2 holds
ex N being Neighbourhood of y0 st
( N c= dom (SVF1 (2,f,u)) & ex L being LinearFunc ex R being RestFunc st
for y being Real st y in N holds
((SVF1 (2,f,u)) . y) - ((SVF1 (2,f,u)) . y0) = (L . (y - y0)) + (R . (y - y0)) )