:: deftheorem Def17 defines hpartdiff32 PDIFF_5:def 17 :
for f being PartFunc of (REAL 3),REAL
for u being Element of REAL 3 st f is_hpartial_differentiable`32_in u holds
for b3 being Real holds
( b3 = hpartdiff32 (f,u) iff ex x0, y0, z0 being Real st
( u = <*x0,y0,z0*> & ex N being Neighbourhood of y0 st
( N c= dom (SVF1 (2,(pdiff1 (f,3)),u)) & ex L being LinearFunc ex R being RestFunc st
( b3 = L . 1 & ( for y being Real st y in N holds
((SVF1 (2,(pdiff1 (f,3)),u)) . y) - ((SVF1 (2,(pdiff1 (f,3)),u)) . y0) = (L . (y - y0)) + (R . (y - y0)) ) ) ) ) );