let u be Element of REAL 3; :: thesis: for f being PartFunc of (REAL 3),REAL holds
( f is_hpartial_differentiable`12_in u iff pdiff1 (f,1) is_partial_differentiable_in u,2 )
let f be PartFunc of (REAL 3),REAL; :: thesis: ( f is_hpartial_differentiable`12_in u iff pdiff1 (f,1) is_partial_differentiable_in u,2 )
thus ( f is_hpartial_differentiable`12_in u implies pdiff1 (f,1) is_partial_differentiable_in u,2 ) :: thesis: ( pdiff1 (f,1) is_partial_differentiable_in u,2 implies f is_hpartial_differentiable`12_in u )

proof

assume f is_hpartial_differentiable`12_in u ; :: thesis: pdiff1 (f,1) is_partial_differentiable_in u,2
then consider x0, y0, z0 being Real such that
A1: ( u = <*x0,y0,z0*> & ex N being Neighbourhood of y0 st
( N c= dom (SVF1 (2,(pdiff1 (f,1)),u)) & ex L being LINEAR ex R being REST st
for y being Real st y in N holds
((SVF1 (2,(pdiff1 (f,1)),u)) . y) - ((SVF1 (2,(pdiff1 (f,1)),u)) . y0) = (L . (y - y0)) + (R . (y - y0)) ) ) by Def2;
thus pdiff1 (f,1) is_partial_differentiable_in u,2 by A1, PDIFF_4:14; :: thesis: verum

end;

assume pdiff1 (f,1) is_partial_differentiable_in u,2 ; :: thesis: f is_hpartial_differentiable`12_in u
then consider x0, y0, z0 being Real such that
A2: ( u = <*x0,y0,z0*> & ex N being Neighbourhood of y0 st
( N c= dom (SVF1 (2,(pdiff1 (f,1)),u)) & ex L being LINEAR ex R being REST st
for y being Real st y in N holds
((SVF1 (2,(pdiff1 (f,1)),u)) . y) - ((SVF1 (2,(pdiff1 (f,1)),u)) . y0) = (L . (y - y0)) + (R . (y - y0)) ) ) by PDIFF_4:14;
thus f is_hpartial_differentiable`12_in u by A2, Def2; :: thesis: verum