let z0 be Element of REAL 2; for f1, f2 being PartFunc of (REAL 2),REAL st f1 is_hpartial_differentiable`11_in z0 & f2 is_hpartial_differentiable`11_in z0 holds
( (pdiff1 (f1,1)) - (pdiff1 (f2,1)) is_partial_differentiable_in z0,1 & partdiff (((pdiff1 (f1,1)) - (pdiff1 (f2,1))),z0,1) = (hpartdiff11 (f1,z0)) - (hpartdiff11 (f2,z0)) )
let f1, f2 be PartFunc of (REAL 2),REAL; ( f1 is_hpartial_differentiable`11_in z0 & f2 is_hpartial_differentiable`11_in z0 implies ( (pdiff1 (f1,1)) - (pdiff1 (f2,1)) is_partial_differentiable_in z0,1 & partdiff (((pdiff1 (f1,1)) - (pdiff1 (f2,1))),z0,1) = (hpartdiff11 (f1,z0)) - (hpartdiff11 (f2,z0)) ) )
assume that
A1:
f1 is_hpartial_differentiable`11_in z0
and
A2:
f2 is_hpartial_differentiable`11_in z0
; ( (pdiff1 (f1,1)) - (pdiff1 (f2,1)) is_partial_differentiable_in z0,1 & partdiff (((pdiff1 (f1,1)) - (pdiff1 (f2,1))),z0,1) = (hpartdiff11 (f1,z0)) - (hpartdiff11 (f2,z0)) )
A3:
( pdiff1 (f1,1) is_partial_differentiable_in z0,1 & pdiff1 (f2,1) is_partial_differentiable_in z0,1 )
by A1, A2, Th9;
then
partdiff (((pdiff1 (f1,1)) - (pdiff1 (f2,1))),z0,1) = (partdiff ((pdiff1 (f1,1)),z0,1)) - (partdiff ((pdiff1 (f2,1)),z0,1))
by PDIFF_1:31;
then partdiff (((pdiff1 (f1,1)) - (pdiff1 (f2,1))),z0,1) =
(hpartdiff11 (f1,z0)) - (partdiff ((pdiff1 (f2,1)),z0,1))
by A1, Th13
.=
(hpartdiff11 (f1,z0)) - (hpartdiff11 (f2,z0))
by A2, Th13
;
hence
( (pdiff1 (f1,1)) - (pdiff1 (f2,1)) is_partial_differentiable_in z0,1 & partdiff (((pdiff1 (f1,1)) - (pdiff1 (f2,1))),z0,1) = (hpartdiff11 (f1,z0)) - (hpartdiff11 (f2,z0)) )
by A3, PDIFF_1:31; verum