let z be Element of REAL 2; :: thesis: for f being PartFunc of (REAL 2),REAL holds
( f is_hpartial_differentiable`21_in z iff pdiff1 f,2 is_partial_differentiable_in z,1 )
let f be PartFunc of (REAL 2),REAL ; :: thesis: ( f is_hpartial_differentiable`21_in z iff pdiff1 f,2 is_partial_differentiable_in z,1 )
thus ( f is_hpartial_differentiable`21_in z implies pdiff1 f,2 is_partial_differentiable_in z,1 ) :: thesis: ( pdiff1 f,2 is_partial_differentiable_in z,1 implies f is_hpartial_differentiable`21_in z )

proof

assume A0: f is_hpartial_differentiable`21_in z ; :: thesis: pdiff1 f,2 is_partial_differentiable_in z,1
consider x0, y0 being Real such that
A1: ( 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 A0, Def5;
thus pdiff1 f,2 is_partial_differentiable_in z,1 by A1, PDIFF_2:9; :: thesis: verum

end;

assume A3: pdiff1 f,2 is_partial_differentiable_in z,1 ; :: thesis: f is_hpartial_differentiable`21_in z
consider x0, y0 being Real such that
A4: ( 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 A3, PDIFF_2:9;
thus f is_hpartial_differentiable`21_in z by A4, Def5; :: thesis: verum