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

proof

assume A0: f is_hpartial_differentiable`12_in z ; :: thesis: pdiff1 f,1 is_partial_differentiable_in z,2
consider x0, y0 being Real such that
A1: ( z = <*x0,y0*> & ex N being Neighbourhood of y0 st
( N c= dom (SVF1 2,(pdiff1 f,1),z) & ex L being LINEAR ex R being REST st
for y being Real st y in N holds
((SVF1 2,(pdiff1 f,1),z) . y) - ((SVF1 2,(pdiff1 f,1),z) . y0) = (L . (y - y0)) + (R . (y - y0)) ) ) by A0, Def4;
thus pdiff1 f,1 is_partial_differentiable_in z,2 by A1, PDIFF_2:10; :: thesis: verum

end;

assume A3: pdiff1 f,1 is_partial_differentiable_in z,2 ; :: thesis: f is_hpartial_differentiable`12_in z
consider x0, y0 being Real such that
A4: ( z = <*x0,y0*> & ex N being Neighbourhood of y0 st
( N c= dom (SVF1 2,(pdiff1 f,1),z) & ex L being LINEAR ex R being REST st
for y being Real st y in N holds
((SVF1 2,(pdiff1 f,1),z) . y) - ((SVF1 2,(pdiff1 f,1),z) . y0) = (L . (y - y0)) + (R . (y - y0)) ) ) by A3, PDIFF_2:10;
thus f is_hpartial_differentiable`12_in z by A4, Def4; :: thesis: verum