thus ( f is_partial_differentiable`2_in z implies ex x0, y0 being Real st
( z = <*x0,y0*> & ex N being Neighbourhood of y0 st
( N c= dom (SVF2 f,z) & ex L being LINEAR ex R being REST st
for y being Real st y in N holds
((SVF2 f,z) . y) - ((SVF2 f,z) . y0) = (L . (y - y0)) + (R . (y - y0)) ) ) ) :: thesis: ( ex x0, y0 being Real st
( z = <*x0,y0*> & ex N being Neighbourhood of y0 st
( N c= dom (SVF2 f,z) & ex L being LINEAR ex R being REST st
for y being Real st y in N holds
((SVF2 f,z) . y) - ((SVF2 f,z) . y0) = (L . (y - y0)) + (R . (y - y0)) ) ) implies f is_partial_differentiable`2_in z )

proof

assume A1: f is_partial_differentiable`2_in z ; :: thesis: ex x0, y0 being Real st
( z = <*x0,y0*> & ex N being Neighbourhood of y0 st
( N c= dom (SVF2 f,z) & ex L being LINEAR ex R being REST st
for y being Real st y in N holds
((SVF2 f,z) . y) - ((SVF2 f,z) . y0) = (L . (y - y0)) + (R . (y - y0)) ) )
consider x0, y0 being Real such that
A2: ( z = <*x0,y0*> & SVF2 f,z is_differentiable_in y0 ) by A1, Def7;
ex N being Neighbourhood of y0 st
( N c= dom (SVF2 f,z) & ex L being LINEAR ex R being REST st
for y being Real st y in N holds
((SVF2 f,z) . y) - ((SVF2 f,z) . y0) = (L . (y - y0)) + (R . (y - y0)) ) by A2, FDIFF_1:def 5;
hence ex x0, y0 being Real st
( z = <*x0,y0*> & ex N being Neighbourhood of y0 st
( N c= dom (SVF2 f,z) & ex L being LINEAR ex R being REST st
for y being Real st y in N holds
((SVF2 f,z) . y) - ((SVF2 f,z) . y0) = (L . (y - y0)) + (R . (y - y0)) ) ) by A2; :: thesis: verum

end;

assume ex x0, y0 being Real st
( z = <*x0,y0*> & ex N being Neighbourhood of y0 st
( N c= dom (SVF2 f,z) & ex L being LINEAR ex R being REST st
for y being Real st y in N holds
((SVF2 f,z) . y) - ((SVF2 f,z) . y0) = (L . (y - y0)) + (R . (y - y0)) ) ) ; :: thesis: f is_partial_differentiable`2_in z
then consider x0, y0 being Real such that
A4: ( z = <*x0,y0*> & ex N being Neighbourhood of y0 st
( N c= dom (SVF2 f,z) & ex L being LINEAR ex R being REST st
for y being Real st y in N holds
((SVF2 f,z) . y) - ((SVF2 f,z) . y0) = (L . (y - y0)) + (R . (y - y0)) ) ) ;
consider N being Neighbourhood of y0 such that
A5: ( N c= dom (SVF2 f,z) & ex L being LINEAR ex R being REST st
for y being Real st y in N holds
((SVF2 f,z) . y) - ((SVF2 f,z) . y0) = (L . (y - y0)) + (R . (y - y0)) ) by A4;
SVF2 f,z is_differentiable_in y0 by A5, FDIFF_1:def 5;
hence f is_partial_differentiable`2_in z by A4, Def7; :: thesis: verum