let u0 be Element of REAL 3; for f1, f2 being PartFunc of (REAL 3),REAL st f1 is_hpartial_differentiable`21_in u0 & f2 is_hpartial_differentiable`21_in u0 holds
( (pdiff1 (f1,2)) + (pdiff1 (f2,2)) is_partial_differentiable_in u0,1 & partdiff (((pdiff1 (f1,2)) + (pdiff1 (f2,2))),u0,1) = (hpartdiff21 (f1,u0)) + (hpartdiff21 (f2,u0)) )
let f1, f2 be PartFunc of (REAL 3),REAL; ( f1 is_hpartial_differentiable`21_in u0 & f2 is_hpartial_differentiable`21_in u0 implies ( (pdiff1 (f1,2)) + (pdiff1 (f2,2)) is_partial_differentiable_in u0,1 & partdiff (((pdiff1 (f1,2)) + (pdiff1 (f2,2))),u0,1) = (hpartdiff21 (f1,u0)) + (hpartdiff21 (f2,u0)) ) )
assume that
A1:
f1 is_hpartial_differentiable`21_in u0
and
A2:
f2 is_hpartial_differentiable`21_in u0
; ( (pdiff1 (f1,2)) + (pdiff1 (f2,2)) is_partial_differentiable_in u0,1 & partdiff (((pdiff1 (f1,2)) + (pdiff1 (f2,2))),u0,1) = (hpartdiff21 (f1,u0)) + (hpartdiff21 (f2,u0)) )
A3:
pdiff1 (f1,2) is_partial_differentiable_in u0,1
by A1, Th22;
A4:
pdiff1 (f2,2) is_partial_differentiable_in u0,1
by A2, Th22;
then
( (pdiff1 (f1,2)) + (pdiff1 (f2,2)) is_partial_differentiable_in u0,1 & partdiff (((pdiff1 (f1,2)) + (pdiff1 (f2,2))),u0,1) = (partdiff ((pdiff1 (f1,2)),u0,1)) + (partdiff ((pdiff1 (f2,2)),u0,1)) )
by A3, PDIFF_1:29;
then partdiff (((pdiff1 (f1,2)) + (pdiff1 (f2,2))),u0,1) =
(hpartdiff21 (f1,u0)) + (partdiff ((pdiff1 (f2,2)),u0,1))
by A1, Th31
.=
(hpartdiff21 (f1,u0)) + (hpartdiff21 (f2,u0))
by A2, Th31
;
hence
( (pdiff1 (f1,2)) + (pdiff1 (f2,2)) is_partial_differentiable_in u0,1 & partdiff (((pdiff1 (f1,2)) + (pdiff1 (f2,2))),u0,1) = (hpartdiff21 (f1,u0)) + (hpartdiff21 (f2,u0)) )
by A3, A4, PDIFF_1:29; verum