theorem Th16: :: PDIFF_4:16

for x0, y0, z0, r 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,1 holds
( r = partdiff (f,u,1) iff ex x0, y0, z0 being Real st
( u = <*x0,y0,z0*> & ex N being Neighbourhood of x0 st
( N c= dom (SVF1 (1,f,u)) & ex L being LinearFunc ex R being RestFunc st
( r = L . 1 & ( for x being Real st x in N holds
((SVF1 (1,f,u)) . x) - ((SVF1 (1,f,u)) . x0) = (L . (x - x0)) + (R . (x - x0)) ) ) ) ) )