let z0 be Element of REAL 2; :: thesis: for f1, f2 being PartFunc of (REAL 2),REAL st f1 is_hpartial_differentiable`12_in z0 & f2 is_hpartial_differentiable`12_in z0 holds
( (pdiff1 (f1,1)) + (pdiff1 (f2,1)) is_partial_differentiable_in z0,2 & partdiff (((pdiff1 (f1,1)) + (pdiff1 (f2,1))),z0,2) = (hpartdiff12 (f1,z0)) + (hpartdiff12 (f2,z0)) )
let f1, f2 be PartFunc of (REAL 2),REAL; :: thesis: ( f1 is_hpartial_differentiable`12_in z0 & f2 is_hpartial_differentiable`12_in z0 implies ( (pdiff1 (f1,1)) + (pdiff1 (f2,1)) is_partial_differentiable_in z0,2 & partdiff (((pdiff1 (f1,1)) + (pdiff1 (f2,1))),z0,2) = (hpartdiff12 (f1,z0)) + (hpartdiff12 (f2,z0)) ) )
assume that
A1: f1 is_hpartial_differentiable`12_in z0 and
A2: f2 is_hpartial_differentiable`12_in z0 ; :: thesis: ( (pdiff1 (f1,1)) + (pdiff1 (f2,1)) is_partial_differentiable_in z0,2 & partdiff (((pdiff1 (f1,1)) + (pdiff1 (f2,1))),z0,2) = (hpartdiff12 (f1,z0)) + (hpartdiff12 (f2,z0)) )
A3: ( pdiff1 (f1,1) is_partial_differentiable_in z0,2 & pdiff1 (f2,1) is_partial_differentiable_in z0,2 ) by A1, A2, Th10;
then partdiff (((pdiff1 (f1,1)) + (pdiff1 (f2,1))),z0,2) = (partdiff ((pdiff1 (f1,1)),z0,2)) + (partdiff ((pdiff1 (f2,1)),z0,2)) by PDIFF_1:29;
then partdiff (((pdiff1 (f1,1)) + (pdiff1 (f2,1))),z0,2) = (hpartdiff12 (f1,z0)) + (partdiff ((pdiff1 (f2,1)),z0,2)) by A1, Th14
.= (hpartdiff12 (f1,z0)) + (hpartdiff12 (f2,z0)) by A2, Th14 ;
hence ( (pdiff1 (f1,1)) + (pdiff1 (f2,1)) is_partial_differentiable_in z0,2 & partdiff (((pdiff1 (f1,1)) + (pdiff1 (f2,1))),z0,2) = (hpartdiff12 (f1,z0)) + (hpartdiff12 (f2,z0)) ) by A3, PDIFF_1:29; :: thesis: verum