theorem Th12: :: PDIFF_2:12

for x0, y0, r being Real
for z being Element of REAL 2
for f being PartFunc of (REAL 2),REAL st z = <*x0,y0*> & f is_partial_differentiable_in z,2 holds
( r = partdiff (f,z,2) iff ex x0, y0 being Real st
( z = <*x0,y0*> & ex N being Neighbourhood of y0 st
( N c= dom (SVF1 (2,f,z)) & ex L being LinearFunc ex R being RestFunc st
( r = L . 1 & ( for y being Real st y in N holds
((SVF1 (2,f,z)) . y) - ((SVF1 (2,f,z)) . y0) = (L . (y - y0)) + (R . (y - y0)) ) ) ) ) )