:: deftheorem Def9 defines hpartdiff22 PDIFF_3:def 9 :
for f being PartFunc of (REAL 2),REAL
for z being Element of REAL 2 st f is_hpartial_differentiable`22_in z holds
for b3 being Real holds
( b3 = hpartdiff22 (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,2)),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,2)),z)) . y) - ((SVF1 (2,(pdiff1 (f,2)),z)) . y0) = (L . (y - y0)) + (R . (y - y0)) ) ) ) ) );