:: deftheorem Def6 defines hpartdiff11 PDIFF_3:def 6 :
for f being PartFunc of (REAL 2),REAL
for z being Element of REAL 2 st f is_hpartial_differentiable`11_in z holds
for b3 being Real holds
( b3 = hpartdiff11 (f,z) iff ex x0, y0 being Real st
( z = <*x0,y0*> & ex N being Neighbourhood of x0 st
( N c= dom (SVF1 (1,(pdiff1 (f,1)),z)) & ex L being LinearFunc ex R being RestFunc st
( b3 = L . 1 & ( for x being Real st x in N holds
((SVF1 (1,(pdiff1 (f,1)),z)) . x) - ((SVF1 (1,(pdiff1 (f,1)),z)) . x0) = (L . (x - x0)) + (R . (x - x0)) ) ) ) ) );