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 f is_hpartial_differentiable`21_in z ; :: thesis: pdiff1 (f,2) is_partial_differentiable_in z,1
then 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
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 Def4;
hence pdiff1 (f,2) is_partial_differentiable_in z,1 by PDIFF_2:9; :: thesis: verum

end;

assume pdiff1 (f,2) is_partial_differentiable_in z,1 ; :: thesis: f is_hpartial_differentiable`21_in z
then 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
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 PDIFF_2:9;
hence f is_hpartial_differentiable`21_in z by Def4; :: thesis: verum