let u0 be Element of REAL 3; :: thesis: for f1, f2 being PartFunc of (REAL 3),REAL st f1 is_hpartial_differentiable`32_in u0 & f2 is_hpartial_differentiable`32_in u0 holds
(pdiff1 f1,3) (#) (pdiff1 f2,3) is_partial_differentiable_in u0,2
let f1, f2 be PartFunc of (REAL 3),REAL ; :: thesis: ( f1 is_hpartial_differentiable`32_in u0 & f2 is_hpartial_differentiable`32_in u0 implies (pdiff1 f1,3) (#) (pdiff1 f2,3) is_partial_differentiable_in u0,2 )
assume ( f1 is_hpartial_differentiable`32_in u0 & f2 is_hpartial_differentiable`32_in u0 ) ; :: thesis: (pdiff1 f1,3) (#) (pdiff1 f2,3) is_partial_differentiable_in u0,2
then ( pdiff1 f1,3 is_partial_differentiable_in u0,2 & pdiff1 f2,3 is_partial_differentiable_in u0,2 ) by Th12ForSecondOrder;
hence (pdiff1 f1,3) (#) (pdiff1 f2,3) is_partial_differentiable_in u0,2 by PDIFF_4:29; :: thesis: verum