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`32_in u holds
hpartdiff32 f,u = diff (SVF1 2,(pdiff1 f,3),u),y0
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`32_in u holds
hpartdiff32 f,u = diff (SVF1 2,(pdiff1 f,3),u),y0
let f be PartFunc of (REAL 3),REAL ; :: thesis: ( u = <*x0,y0,z0*> & f is_hpartial_differentiable`32_in u implies hpartdiff32 f,u = diff (SVF1 2,(pdiff1 f,3),u),y0 )
set r = hpartdiff32 f,u;
assume that
A1: u = <*x0,y0,z0*> and
A2: f is_hpartial_differentiable`32_in u ; :: thesis: hpartdiff32 f,u = diff (SVF1 2,(pdiff1 f,3),u),y0
consider x1, y1, z1 being Real such that
A3: ( u = <*x1,y1,z1*> & ex N being Neighbourhood of y1 st
( N c= dom (SVF1 2,(pdiff1 f,3),u) & ex L being LINEAR ex R being REST st
for y being Real st y in N holds
((SVF1 2,(pdiff1 f,3),u) . y) - ((SVF1 2,(pdiff1 f,3),u) . y1) = (L . (y - y1)) + (R . (y - y1)) ) ) by A2, Def6ForSecondOrder;
consider N being Neighbourhood of y1 such that
A4: ( N c= dom (SVF1 2,(pdiff1 f,3),u) & ex L being LINEAR ex R being REST st
for y being Real st y in N holds
((SVF1 2,(pdiff1 f,3),u) . y) - ((SVF1 2,(pdiff1 f,3),u) . y1) = (L . (y - y1)) + (R . (y - y1)) ) by A3;
consider L being LINEAR, R being REST such that
A5: for y being Real st y in N holds
((SVF1 2,(pdiff1 f,3),u) . y) - ((SVF1 2,(pdiff1 f,3),u) . y1) = (L . (y - y1)) + (R . (y - y1)) by A4;
A6: ( x0 = x1 & y0 = y1 & z0 = z1 ) by A1, A3, FINSEQ_1:99;
A7: hpartdiff32 f,u = L . 1 by A2, A3, A4, A5, Def10ForSecondOrder;
SVF1 2,(pdiff1 f,3),u is_differentiable_in y0 by A4, A6, FDIFF_1:def 5;
hence hpartdiff32 f,u = diff (SVF1 2,(pdiff1 f,3),u),y0 by A4, A5, A6, A7, FDIFF_1:def 6; :: thesis: verum