consider x0, y0, z0 being Real such that
A2: ( u = <*x0,y0,z0*> & ex N being Neighbourhood of y0 st
( N c= dom (SVF1 (2,(pdiff1 (f,2)),u)) & ex L being LinearFunc ex R being RestFunc st
for y being Real st y in N holds
((SVF1 (2,(pdiff1 (f,2)),u)) . y) - ((SVF1 (2,(pdiff1 (f,2)),u)) . y0) = (L . (y - y0)) + (R . (y - y0)) ) ) by A1;
consider N being Neighbourhood of y0 such that
A3: ( N c= dom (SVF1 (2,(pdiff1 (f,2)),u)) & ex L being LinearFunc ex R being RestFunc st
for y being Real st y in N holds
((SVF1 (2,(pdiff1 (f,2)),u)) . y) - ((SVF1 (2,(pdiff1 (f,2)),u)) . y0) = (L . (y - y0)) + (R . (y - y0)) ) by A2;
consider L being LinearFunc, R being RestFunc such that
A4: for y being Real st y in N holds
((SVF1 (2,(pdiff1 (f,2)),u)) . y) - ((SVF1 (2,(pdiff1 (f,2)),u)) . y0) = (L . (y - y0)) + (R . (y - y0)) 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 ; :: thesis: ex x0, y0, z0 being Real st
( u = <*x0,y0,z0*> & ex N being Neighbourhood of y0 st
( N c= dom (SVF1 (2,(pdiff1 (f,2)),u)) & 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,2)),u)) . y) - ((SVF1 (2,(pdiff1 (f,2)),u)) . y0) = (L . (y - y0)) + (R . (y - y0)) ) ) ) )
L . 1 = r * 1 by A5
.= r ;
hence ex x0, y0, z0 being Real st
( u = <*x0,y0,z0*> & ex N being Neighbourhood of y0 st
( N c= dom (SVF1 (2,(pdiff1 (f,2)),u)) & 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,2)),u)) . y) - ((SVF1 (2,(pdiff1 (f,2)),u)) . y0) = (L . (y - y0)) + (R . (y - y0)) ) ) ) ) by A2, A3, A4; :: thesis: verum