let z0 be Element of REAL 2; :: thesis: for f being PartFunc of REAL 2, REAL st f is_hpartial_differentiable`11_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`11_in z0 implies ex R being REST st
( R . 0 = 0 & R is_continuous_in 0 ) )
assume f is_hpartial_differentiable`11_in z0 ; :: thesis: ex R being REST st
( R . 0 = 0 & R is_continuous_in 0 )
then pdiff1 f,z0 is_partial_differentiable`1_in z0 by Th9;
hence ex R being REST st
( R . 0 = 0 & R is_continuous_in 0 ) by PDIFF_2:27; :: thesis: verum