consider x0, y0, z0 being Real such that
A2:
( u = <*x0,y0,z0*> & ex N being Neighbourhood of z0 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)) . z0) = (L . (z - z0)) + (R . (z - z0)) ) )
by A1;
consider N being Neighbourhood of z0 such that
A3:
( 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)) . z0) = (L . (z - z0)) + (R . (z - z0)) )
by A2;
consider L being LinearFunc, R being RestFunc such that
A4:
for z being Real st z in N holds
((SVF1 (3,(pdiff1 (f,3)),u)) . z) - ((SVF1 (3,(pdiff1 (f,3)),u)) . z0) = (L . (z - z0)) + (R . (z - z0))
by A3;
consider r being Real such that
A5:
for p being Real holds L . p = r * p
by FDIFF_1:def 3;
take
r
; ex x0, y0, z0 being Real st
( u = <*x0,y0,z0*> & ex N being Neighbourhood of z0 st
( N c= dom (SVF1 (3,(pdiff1 (f,3)),u)) & ex L being LinearFunc ex R being RestFunc st
( r = L . 1 & ( for z being Real st z in N holds
((SVF1 (3,(pdiff1 (f,3)),u)) . z) - ((SVF1 (3,(pdiff1 (f,3)),u)) . z0) = (L . (z - z0)) + (R . (z - z0)) ) ) ) )
L . 1 =
r * 1
by A5
.=
r
;
hence
ex x0, y0, z0 being Real st
( u = <*x0,y0,z0*> & ex N being Neighbourhood of z0 st
( N c= dom (SVF1 (3,(pdiff1 (f,3)),u)) & ex L being LinearFunc ex R being RestFunc st
( r = L . 1 & ( for z being Real st z in N holds
((SVF1 (3,(pdiff1 (f,3)),u)) . z) - ((SVF1 (3,(pdiff1 (f,3)),u)) . z0) = (L . (z - z0)) + (R . (z - z0)) ) ) ) )
by A2, A3, A4; verum