let u0 be Element of REAL 3; for f1, f2 being PartFunc of (REAL 3),REAL st f1 is_hpartial_differentiable`13_in u0 & f2 is_hpartial_differentiable`13_in u0 holds
(pdiff1 (f1,1)) (#) (pdiff1 (f2,1)) is_partial_differentiable_in u0,3
let f1, f2 be PartFunc of (REAL 3),REAL; ( f1 is_hpartial_differentiable`13_in u0 & f2 is_hpartial_differentiable`13_in u0 implies (pdiff1 (f1,1)) (#) (pdiff1 (f2,1)) is_partial_differentiable_in u0,3 )
assume
( f1 is_hpartial_differentiable`13_in u0 & f2 is_hpartial_differentiable`13_in u0 )
; (pdiff1 (f1,1)) (#) (pdiff1 (f2,1)) is_partial_differentiable_in u0,3
then
( pdiff1 (f1,1) is_partial_differentiable_in u0,3 & pdiff1 (f2,1) is_partial_differentiable_in u0,3 )
by Th21;
hence
(pdiff1 (f1,1)) (#) (pdiff1 (f2,1)) is_partial_differentiable_in u0,3
by PDIFF_4:30; verum