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