let u0 be Element of REAL 3; :: thesis: for f being PartFunc of (REAL 3),REAL st f is_hpartial_differentiable`22_in u0 holds
ex R being REST st
( R . 0 = 0 & R is_continuous_in 0 )
let f be PartFunc of (REAL 3),REAL ; :: thesis: ( f is_hpartial_differentiable`22_in u0 implies ex R being REST st
( R . 0 = 0 & R is_continuous_in 0 ) )
assume f is_hpartial_differentiable`22_in u0 ; :: thesis: ex R being REST st
( R . 0 = 0 & R is_continuous_in 0 )
then pdiff1 f,2 is_partial_differentiable_in u0,2 by Th12;
hence ex R being REST st
( R . 0 = 0 & R is_continuous_in 0 ) by PDIFF_4:35; :: thesis: verum