consider x0, y0 being Real such that
A2:
z = <*x0,y0*>
and
A3:
ex N being Neighbourhood of y0 st
( N c= dom (SVF1 (2,(pdiff1 (f,1)),z)) & ex L being LinearFunc ex R being RestFunc st
for y being Real st y in N holds
((SVF1 (2,(pdiff1 (f,1)),z)) . y) - ((SVF1 (2,(pdiff1 (f,1)),z)) . y0) = (L . (y - y0)) + (R . (y - y0)) )
by A1;
consider N being Neighbourhood of y0 such that
A4:
N c= dom (SVF1 (2,(pdiff1 (f,1)),z))
and
A5:
ex L being LinearFunc ex R being RestFunc st
for y being Real st y in N holds
((SVF1 (2,(pdiff1 (f,1)),z)) . y) - ((SVF1 (2,(pdiff1 (f,1)),z)) . y0) = (L . (y - y0)) + (R . (y - y0))
by A3;
consider L being LinearFunc, R being RestFunc such that
A6:
for y being Real st y in N holds
((SVF1 (2,(pdiff1 (f,1)),z)) . y) - ((SVF1 (2,(pdiff1 (f,1)),z)) . y0) = (L . (y - y0)) + (R . (y - y0))
by A5;
consider r being Real such that
A7:
for p being Real holds L . p = r * p
by FDIFF_1:def 3;
take
r
; ex x0, y0 being Real st
( z = <*x0,y0*> & ex N being Neighbourhood of y0 st
( N c= dom (SVF1 (2,(pdiff1 (f,1)),z)) & ex L being LinearFunc ex R being RestFunc st
( r = L . 1 & ( for y being Real st y in N holds
((SVF1 (2,(pdiff1 (f,1)),z)) . y) - ((SVF1 (2,(pdiff1 (f,1)),z)) . y0) = (L . (y - y0)) + (R . (y - y0)) ) ) ) )
L . 1 =
r * 1
by A7
.=
r
;
hence
ex x0, y0 being Real st
( z = <*x0,y0*> & ex N being Neighbourhood of y0 st
( N c= dom (SVF1 (2,(pdiff1 (f,1)),z)) & ex L being LinearFunc ex R being RestFunc st
( r = L . 1 & ( for y being Real st y in N holds
((SVF1 (2,(pdiff1 (f,1)),z)) . y) - ((SVF1 (2,(pdiff1 (f,1)),z)) . y0) = (L . (y - y0)) + (R . (y - y0)) ) ) ) )
by A2, A4, A6; verum