consider x0, y0 being Real such that
A2: ( z = <*x0,y0*> & ex N being Neighbourhood of x0 st
( N c= dom (SVF1 1,(pdiff1 f,2),z) & ex L being LINEAR ex R being REST st
for x being Real st x in N holds
((SVF1 1,(pdiff1 f,2),z) . x) - ((SVF1 1,(pdiff1 f,2),z) . x0) = (L . (x - x0)) + (R . (x - x0)) ) ) by A1, Def5;
consider N being Neighbourhood of x0 such that
A3: ( N c= dom (SVF1 1,(pdiff1 f,2),z) & ex L being LINEAR ex R being REST st
for x being Real st x in N holds
((SVF1 1,(pdiff1 f,2),z) . x) - ((SVF1 1,(pdiff1 f,2),z) . x0) = (L . (x - x0)) + (R . (x - x0)) ) by A2;
consider L being LINEAR, R being REST such that
A4: for x being Real st x in N holds
((SVF1 1,(pdiff1 f,2),z) . x) - ((SVF1 1,(pdiff1 f,2),z) . x0) = (L . (x - x0)) + (R . (x - x0)) by A3;
consider r being Real such that
A5: for p being Real holds L . p = r * p by FDIFF_1:def 4;
take r ; :: thesis: ex x0, y0 being Real st
( z = <*x0,y0*> & ex N being Neighbourhood of x0 st
( N c= dom (SVF1 1,(pdiff1 f,2),z) & ex L being LINEAR ex R being REST st
( r = L . 1 & ( for x being Real st x in N holds
((SVF1 1,(pdiff1 f,2),z) . x) - ((SVF1 1,(pdiff1 f,2),z) . x0) = (L . (x - x0)) + (R . (x - x0)) ) ) ) )
L . 1 = r * 1 by A5
.= r ;
hence ex x0, y0 being Real st
( z = <*x0,y0*> & ex N being Neighbourhood of x0 st
( N c= dom (SVF1 1,(pdiff1 f,2),z) & ex L being LINEAR ex R being REST st
( r = L . 1 & ( for x being Real st x in N holds
((SVF1 1,(pdiff1 f,2),z) . x) - ((SVF1 1,(pdiff1 f,2),z) . x0) = (L . (x - x0)) + (R . (x - x0)) ) ) ) ) by A2, A3, A4; :: thesis: verum