let z0 be Element of REAL 2; :: thesis: for f being PartFunc of (REAL 2),REAL st f is_hpartial_differentiable`21_in z0 holds
ex R being REST st
( R . 0 = 0 & R is_continuous_in 0 )
let f be PartFunc of (REAL 2),REAL ; :: thesis: ( f is_hpartial_differentiable`21_in z0 implies ex R being REST st
( R . 0 = 0 & R is_continuous_in 0 ) )
assume f is_hpartial_differentiable`21_in z0 ; :: thesis: ex R being REST st
( R . 0 = 0 & R is_continuous_in 0 )
then pdiff1 f,2 is_partial_differentiable_in z0,1 by Th11;
hence ex R being REST st
( R . 0 = 0 & R is_continuous_in 0 ) by PDIFF_2:23; :: thesis: verum