let f be PartFunc of (REAL 2),REAL ; :: thesis: for z being Element of REAL 2 holds
( f is_partial_differentiable_in z,1 iff ex x0, y0 being Real st
( z = <*x0,y0*> & ex N being Neighbourhood of x0 st
( N c= dom (SVF1 1,f,z) & ex L being LINEAR ex R being REST st
for x being Real st x in N holds
((SVF1 1,f,z) . x) - ((SVF1 1,f,z) . x0) = (L . (x - x0)) + (R . (x - x0)) ) ) )
let z be Element of REAL 2; :: thesis: ( f is_partial_differentiable_in z,1 iff ex x0, y0 being Real st
( z = <*x0,y0*> & ex N being Neighbourhood of x0 st
( N c= dom (SVF1 1,f,z) & ex L being LINEAR ex R being REST st
for x being Real st x in N holds
((SVF1 1,f,z) . x) - ((SVF1 1,f,z) . x0) = (L . (x - x0)) + (R . (x - x0)) ) ) )

hereby :: thesis: ( ex x0, y0 being Real st
( z = <*x0,y0*> & ex N being Neighbourhood of x0 st
( N c= dom (SVF1 1,f,z) & ex L being LINEAR ex R being REST st
for x being Real st x in N holds
((SVF1 1,f,z) . x) - ((SVF1 1,f,z) . x0) = (L . (x - x0)) + (R . (x - x0)) ) ) implies f is_partial_differentiable_in z,1 )

assume A1: f is_partial_differentiable_in z,1 ; :: thesis: ex x0, y0 being Real st
( z = <*x0,y0*> & ex N being Neighbourhood of x0 st
( N c= dom (SVF1 1,f,z) & ex L being LINEAR ex R being REST st
for x being Real st x in N holds
((SVF1 1,f,z) . x) - ((SVF1 1,f,z) . x0) = (L . (x - x0)) + (R . (x - x0)) ) )
thus ex x0, y0 being Real st
( z = <*x0,y0*> & ex N being Neighbourhood of x0 st
( N c= dom (SVF1 1,f,z) & ex L being LINEAR ex R being REST st
for x being Real st x in N holds
((SVF1 1,f,z) . x) - ((SVF1 1,f,z) . x0) = (L . (x - x0)) + (R . (x - x0)) ) ) :: thesis: verum

proof

consider x0, y0 being Real such that
A2: ( z = <*x0,y0*> & SVF1 1,f,z is_differentiable_in x0 ) by A1, Def6;
ex N being Neighbourhood of x0 st
( N c= dom (SVF1 1,f,z) & ex L being LINEAR ex R being REST st
for x being Real st x in N holds
((SVF1 1,f,z) . x) - ((SVF1 1,f,z) . x0) = (L . (x - x0)) + (R . (x - x0)) ) by A2, FDIFF_1:def 5;
hence ex x0, y0 being Real st
( z = <*x0,y0*> & ex N being Neighbourhood of x0 st
( N c= dom (SVF1 1,f,z) & ex L being LINEAR ex R being REST st
for x being Real st x in N holds
((SVF1 1,f,z) . x) - ((SVF1 1,f,z) . x0) = (L . (x - x0)) + (R . (x - x0)) ) ) by A2; :: thesis: verum

end;

end;

assume ex x0, y0 being Real st
( z = <*x0,y0*> & ex N being Neighbourhood of x0 st
( N c= dom (SVF1 1,f,z) & ex L being LINEAR ex R being REST st
for x being Real st x in N holds
((SVF1 1,f,z) . x) - ((SVF1 1,f,z) . x0) = (L . (x - x0)) + (R . (x - x0)) ) ) ; :: thesis: f is_partial_differentiable_in z,1
then consider x0, y0 being Real such that
A4: ( z = <*x0,y0*> & ex N being Neighbourhood of x0 st
( N c= dom (SVF1 1,f,z) & ex L being LINEAR ex R being REST st
for x being Real st x in N holds
((SVF1 1,f,z) . x) - ((SVF1 1,f,z) . x0) = (L . (x - x0)) + (R . (x - x0)) ) ) ;
consider N being Neighbourhood of x0 such that
A5: ( N c= dom (SVF1 1,f,z) & ex L being LINEAR ex R being REST st
for x being Real st x in N holds
((SVF1 1,f,z) . x) - ((SVF1 1,f,z) . x0) = (L . (x - x0)) + (R . (x - x0)) ) by A4;
SVF1 1,f,z is_differentiable_in x0 by A5, FDIFF_1:def 5;
hence f is_partial_differentiable_in z,1 by A4, Def6; :: thesis: verum