:: deftheorem Def7 defines hpartdiff12 PDIFF_3:def 7 :
for f being PartFunc of (REAL 2),REAL
for z being Element of REAL 2 st f is_hpartial_differentiable`12_in z holds
for b₃ being Real holds
( b₃ = hpartdiff12 (f,z) iff ex x0, y0 being Real st
( z = <*x0,y0*> & ex N being Neighbourhood of y0 st
( N c= dom (SVF1 (2,(pdiff1 (f,1)),z)) & 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,1)),z)) . y) - ((SVF1 (2,(pdiff1 (f,1)),z)) . y0) = (L . (y - y0)) + (R . (y - y0)) ) ) ) ) );