let z be Element of REAL 2; :: thesis: for f being PartFunc of (REAL 2),REAL holds
( f is_hpartial_differentiable`22_in z iff pdiff1 (f,2) is_partial_differentiable_in z,2 )
let f be PartFunc of (REAL 2),REAL; :: thesis: ( f is_hpartial_differentiable`22_in z iff pdiff1 (f,2) is_partial_differentiable_in z,2 )
thus ( f is_hpartial_differentiable`22_in z implies pdiff1 (f,2) is_partial_differentiable_in z,2 ) :: thesis: ( pdiff1 (f,2) is_partial_differentiable_in z,2 implies f is_hpartial_differentiable`22_in z )

proof

assume f is_hpartial_differentiable`22_in z ; :: thesis: pdiff1 (f,2) is_partial_differentiable_in z,2
then ex x0, y0 being Real st
( z = <*x0,y0*> & ex N being Neighbourhood of y0 st
( N c= dom (SVF1 (2,(pdiff1 (f,2)),z)) & ex L being LINEAR ex R being REST st
for y being Real st y in N holds
((SVF1 (2,(pdiff1 (f,2)),z)) . y) - ((SVF1 (2,(pdiff1 (f,2)),z)) . y0) = (L . (y - y0)) + (R . (y - y0)) ) ) by Def5;
hence pdiff1 (f,2) is_partial_differentiable_in z,2 by PDIFF_2:10; :: thesis: verum

end;

assume pdiff1 (f,2) is_partial_differentiable_in z,2 ; :: thesis: f is_hpartial_differentiable`22_in z
then ex x0, y0 being Real st
( z = <*x0,y0*> & ex N being Neighbourhood of y0 st
( N c= dom (SVF1 (2,(pdiff1 (f,2)),z)) & ex L being LINEAR ex R being REST st
for y being Real st y in N holds
((SVF1 (2,(pdiff1 (f,2)),z)) . y) - ((SVF1 (2,(pdiff1 (f,2)),z)) . y0) = (L . (y - y0)) + (R . (y - y0)) ) ) by PDIFF_2:10;
hence f is_hpartial_differentiable`22_in z by Def5; :: thesis: verum