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
hpartdiff11 (f,z) = diff ((SVF1 (1,(pdiff1 (f,1)),z)),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
hpartdiff11 (f,z) = diff ((SVF1 (1,(pdiff1 (f,1)),z)),x0)
let f be PartFunc of (REAL 2),REAL; :: thesis: ( z = <*x0,y0*> & f is_hpartial_differentiable`11_in z implies hpartdiff11 (f,z) = diff ((SVF1 (1,(pdiff1 (f,1)),z)),x0) )
set r = hpartdiff11 (f,z);
assume that
A1: z = <*x0,y0*> and
A2: f is_hpartial_differentiable`11_in z ; :: thesis: hpartdiff11 (f,z) = diff ((SVF1 (1,(pdiff1 (f,1)),z)),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 LinearFunc ex R being RestFunc 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;
consider N being Neighbourhood of x1 such that
A5: N c= dom (SVF1 (1,(pdiff1 (f,1)),z)) and
A6: ex L being LinearFunc ex R being RestFunc 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 A4;
A7: x0 = x1 by A1, A3, FINSEQ_1:77;
then A8: SVF1 (1,(pdiff1 (f,1)),z) is_differentiable_in x0 by A5, A6, FDIFF_1:def 4;
consider L being LinearFunc, R being RestFunc such that
A9: 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 A6;
hpartdiff11 (f,z) = L . 1 by A2, A3, A5, A9, Def6;
hence hpartdiff11 (f,z) = diff ((SVF1 (1,(pdiff1 (f,1)),z)),x0) by A5, A9, A7, A8, FDIFF_1:def 5; :: thesis: verum