let x0, y0, z0 be Real; :: thesis: for u being Element of REAL 3
for f being PartFunc of (REAL 3),REAL st u = <*x0,y0,z0*> & f is_hpartial_differentiable`23_in u holds
SVF1 (3,(pdiff1 (f,2)),u) is_differentiable_in z0
let u be Element of REAL 3; :: thesis: for f being PartFunc of (REAL 3),REAL st u = <*x0,y0,z0*> & f is_hpartial_differentiable`23_in u holds
SVF1 (3,(pdiff1 (f,2)),u) is_differentiable_in z0
let f be PartFunc of (REAL 3),REAL; :: thesis: ( u = <*x0,y0,z0*> & f is_hpartial_differentiable`23_in u implies SVF1 (3,(pdiff1 (f,2)),u) is_differentiable_in z0 )
assume that
A1: u = <*x0,y0,z0*> and
A2: f is_hpartial_differentiable`23_in u ; :: thesis: SVF1 (3,(pdiff1 (f,2)),u) is_differentiable_in z0
consider x1, y1, z1 being Real such that
A3: ( u = <*x1,y1,z1*> & ex N being Neighbourhood of z1 st
( N c= dom (SVF1 (3,(pdiff1 (f,2)),u)) & ex L being LinearFunc ex R being RestFunc st
for z being Real st z in N holds
((SVF1 (3,(pdiff1 (f,2)),u)) . z) - ((SVF1 (3,(pdiff1 (f,2)),u)) . z1) = (L . (z - z1)) + (R . (z - z1)) ) ) by A2;
z0 = z1 by A1, A3, FINSEQ_1:78;
hence SVF1 (3,(pdiff1 (f,2)),u) is_differentiable_in z0 by A3, FDIFF_1:def 4; :: thesis: verum