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 LinearFunc ex R being RestFunc 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;
consider N being Neighbourhood of z1 such that
A4: ( N c= dom (SVF1 (3,(pdiff1 (f,3)),u)) & ex L being LinearFunc ex R being RestFunc 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 LinearFunc, R being RestFunc 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_1:78;
A7: hpartdiff33 (f,u) = L . 1 by A2, A3, A4, A5, Def18;
SVF1 (3,(pdiff1 (f,3)),u) is_differentiable_in z0 by A4, A6, FDIFF_1:def 4;
hence hpartdiff33 (f,u) = diff ((SVF1 (3,(pdiff1 (f,3)),u)),z0) by A4, A5, A6, A7, FDIFF_1:def 5; :: thesis: verum