let u be Element of REAL 3; :: thesis: for f being PartFunc of (REAL 3),REAL holds
( f is_hpartial_differentiable`33_in u iff pdiff1 f,3 is_partial_differentiable_in u,3 )
let f be PartFunc of (REAL 3),REAL ; :: thesis: ( f is_hpartial_differentiable`33_in u iff pdiff1 f,3 is_partial_differentiable_in u,3 )
thus ( f is_hpartial_differentiable`33_in u implies pdiff1 f,3 is_partial_differentiable_in u,3 ) :: thesis: ( pdiff1 f,3 is_partial_differentiable_in u,3 implies f is_hpartial_differentiable`33_in u )

proof

assume f is_hpartial_differentiable`33_in u ; :: thesis: pdiff1 f,3 is_partial_differentiable_in u,3
then consider x0, y0, z0 being Real such that
A1: ( u = <*x0,y0,z0*> & ex N being Neighbourhood of z0 st
( N c= dom (SVF1 3,(pdiff1 f,3),u) & ex L being LINEAR ex R being REST st
for z being Real st z in N holds
((SVF1 3,(pdiff1 f,3),u) . z) - ((SVF1 3,(pdiff1 f,3),u) . z0) = (L . (z - z0)) + (R . (z - z0)) ) ) by Def6ForZForSecondOrder;
thus pdiff1 f,3 is_partial_differentiable_in u,3 by A1, PDIFF_4:15; :: thesis: verum

end;

assume pdiff1 f,3 is_partial_differentiable_in u,3 ; :: thesis: f is_hpartial_differentiable`33_in u
then consider x0, y0, z0 being Real such that
A4: ( u = <*x0,y0,z0*> & ex N being Neighbourhood of z0 st
( N c= dom (SVF1 3,(pdiff1 f,3),u) & ex L being LINEAR ex R being REST st
for z being Real st z in N holds
((SVF1 3,(pdiff1 f,3),u) . z) - ((SVF1 3,(pdiff1 f,3),u) . z0) = (L . (z - z0)) + (R . (z - z0)) ) ) by PDIFF_4:15;
thus f is_hpartial_differentiable`33_in u by A4, Def6ForZForSecondOrder; :: thesis: verum