let x0, y0 be Real; :: thesis: for z being Element of REAL 2
for f being PartFunc of (REAL 2),REAL st z = <*x0,y0*> & f is_hpartial_differentiable`11_in z holds
SVF1 1,(pdiff1 f,1),z is_differentiable_in x0
let z be Element of REAL 2; :: thesis: for f being PartFunc of (REAL 2),REAL st z = <*x0,y0*> & f is_hpartial_differentiable`11_in z holds
SVF1 1,(pdiff1 f,1),z is_differentiable_in x0
let f be PartFunc of (REAL 2),REAL ; :: thesis: ( z = <*x0,y0*> & f is_hpartial_differentiable`11_in z implies SVF1 1,(pdiff1 f,1),z is_differentiable_in x0 )
assume that
A1: z = <*x0,y0*> and
A2: f is_hpartial_differentiable`11_in z ; :: thesis: SVF1 1,(pdiff1 f,1),z is_differentiable_in x0
consider x1, y1 being Real such that
A3: z = <*x1,y1*> and
A4: ex N being Neighbourhood of x1 st
( N c= dom (SVF1 1,(pdiff1 f,1),z) & ex L being LINEAR ex R being REST st
for x being Real st x in N holds
((SVF1 1,(pdiff1 f,1),z) . x) - ((SVF1 1,(pdiff1 f,1),z) . x1) = (L . (x - x1)) + (R . (x - x1)) ) by A2, Def2;
x0 = x1 by A1, A3, FINSEQ_1:98;
hence SVF1 1,(pdiff1 f,1),z is_differentiable_in x0 by A4, FDIFF_1:def 5; :: thesis: verum