let u be Element of REAL 3; :: thesis: for f being PartFunc of (REAL 3),REAL holds
( f is_hpartial_differentiable`31_in u iff pdiff1 f,3 is_partial_differentiable_in u,1 )
let f be PartFunc of (REAL 3),REAL ; :: thesis: ( f is_hpartial_differentiable`31_in u iff pdiff1 f,3 is_partial_differentiable_in u,1 )
thus ( f is_hpartial_differentiable`31_in u implies pdiff1 f,3 is_partial_differentiable_in u,1 ) :: thesis: ( pdiff1 f,3 is_partial_differentiable_in u,1 implies f is_hpartial_differentiable`31_in u )

proof

assume f is_hpartial_differentiable`31_in u ; :: thesis: pdiff1 f,3 is_partial_differentiable_in u,1
then consider x0, y0, z0 being Real such that
A1: ( u = <*x0,y0,z0*> & ex N being Neighbourhood of x0 st
( N c= dom (SVF1 1,(pdiff1 f,3),u) & ex L being LINEAR ex R being REST st
for x being Real st x in N holds
((SVF1 1,(pdiff1 f,3),u) . x) - ((SVF1 1,(pdiff1 f,3),u) . x0) = (L . (x - x0)) + (R . (x - x0)) ) ) by Def5ForSecondOrder;
thus pdiff1 f,3 is_partial_differentiable_in u,1 by A1, PDIFF_4:13; :: thesis: verum

end;

assume pdiff1 f,3 is_partial_differentiable_in u,1 ; :: thesis: f is_hpartial_differentiable`31_in u
then consider x0, y0, z0 being Real such that
A4: ( u = <*x0,y0,z0*> & ex N being Neighbourhood of x0 st
( N c= dom (SVF1 1,(pdiff1 f,3),u) & ex L being LINEAR ex R being REST st
for x being Real st x in N holds
((SVF1 1,(pdiff1 f,3),u) . x) - ((SVF1 1,(pdiff1 f,3),u) . x0) = (L . (x - x0)) + (R . (x - x0)) ) ) by PDIFF_4:13;
thus f is_hpartial_differentiable`31_in u by A4, Def5ForSecondOrder; :: thesis: verum