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`33_in u holds
hpartdiff33 f,u = diff (SVF1 3,(pdiff1 f,3),u),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`33_in u holds
hpartdiff33 f,u = diff (SVF1 3,(pdiff1 f,3),u),z0
let f be PartFunc of (REAL 3),REAL ; :: thesis: ( u = <*x0,y0,z0*> & f is_hpartial_differentiable`33_in u implies hpartdiff33 f,u = diff (SVF1 3,(pdiff1 f,3),u),z0 )
set r = hpartdiff33 f,u;
assume that
A1: u = <*x0,y0,z0*> and
A2: f is_hpartial_differentiable`33_in u ; :: thesis: hpartdiff33 f,u = diff (SVF1 3,(pdiff1 f,3),u),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,3),u) & ex L being LINEAR ex R being REST st
for z being Real st z in N holds
((SVF1 3,(pdiff1 f,3),u) . z) - ((SVF1 3,(pdiff1 f,3),u) . z1) = (L . (z - z1)) + (R . (z - z1)) ) ) by A2, Def6ForZForSecondOrder;
consider N being Neighbourhood of z1 such that
A4: ( N c= dom (SVF1 3,(pdiff1 f,3),u) & ex L being LINEAR ex R being REST st
for z being Real st z in N holds
((SVF1 3,(pdiff1 f,3),u) . z) - ((SVF1 3,(pdiff1 f,3),u) . z1) = (L . (z - z1)) + (R . (z - z1)) ) by A3;
consider L being LINEAR, R being REST such that
A5: for z being Real st z in N holds
((SVF1 3,(pdiff1 f,3),u) . z) - ((SVF1 3,(pdiff1 f,3),u) . z1) = (L . (z - z1)) + (R . (z - z1)) by A4;
A6: ( x0 = x1 & y0 = y1 & z0 = z1 ) by A1, A3, FINSEQ_1:99;
A7: hpartdiff33 f,u = L . 1 by A2, A3, A4, A5, Def10ForZForSecondOrder;
SVF1 3,(pdiff1 f,3),u is_differentiable_in z0 by A4, A6, FDIFF_1:def 5;
hence hpartdiff33 f,u = diff (SVF1 3,(pdiff1 f,3),u),z0 by A4, A5, A6, A7, FDIFF_1:def 6; :: thesis: verum